### Algorithms and Complexity Seminar

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A Natural Dynamics for Bargaining on Exchange Networks

### Yashodhan Kanoria (Stanford University)

Bargaining networks model the behavior of a set of players that need to reach pairwise agreements for making profits. Nash bargaining solutions are special outcomes of such games that are both stable and balanced. Kleinberg and Tardos proved a sharp algorithmic characterization of such outcomes, but left open the problem of how the actual bargaining process converges to them. A partial answer was provided by Azar et al. who proposed a distributed, but not natural, algorithm for constructing Nash bargaining solutions, without polynomial bounds on its convergence rate. We introduce
a simple and natural model for the bargaining process, and study its convergence rate to Nash bargaining solutions.

At each time step, each player proposes a deal to each of her neighbors. The proposal consists of a share of the potential profit in case of agreement. The share is chosen to be balanced in Nash's sense as far as this is feasible (with respect to the current best alternatives for both players). We prove that, whenever the Nash bargaining solution is unique (and satisfies a positive gap condition) this dynamics converges to it in polynomial time. Our analysis is based on an approximate decoupling phenomenon between the dynamics on different substructures of the network. This is a novel approach to the analysis of a local algorithm on a network.

This is joint work with Mohsen Bayati, Christian Borgs, Jennifer Chayes and Andrea Montanari.