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arXiv · 1812.04611

Fast Algorithms for Rank-1 Bimatrix Games

Bharat Adsul·Jugal Garg·Ruta Mehta·Milind Sohoni·Bernhard von Stengel
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Abstract

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component.

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Bharat Adsul, Jugal Garg, Ruta Mehta, Milind Sohoni, Bernhard von Stengel. 2018-12-11. Fast Algorithms for Rank-1 Bimatrix Games. https://doi.org/10.1287/opre.2020.1981

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