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\lecture{2}{September 13, 2011}{Joe Scriber}
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\section{Preliminaries}
A two-player game is formally defined as follows.
\begin{definition}
A {\em 2-player game} is defined by a pair of $m \times n$ payoff matrices $(R, C)$, whose rows correspond to the strategies of one of the players of the game, called the {\em row player}, and whose columns correspond to the strategies of the other player, called the {\em column player}. The strategy sets of the row and column players are identified respectively with the sets $[m]:=\{1,\ldots,m\}$ and $[n]:=\{1,\ldots,n\}$.
\end{definition}
\noindent The following theorem was established by John Nash in 1950~\cite{Na1}.
\begin{theorem}
Every game has a Nash equilibrium.
\end{theorem}
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\bibitem{Na1} J.~Nash.
\newblock Equilibrium Points in $n$-Person Games.
\newblock {\em Proceedings of the National Academy of Sciences}, 36(1):48--49, 1950.
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