Fast Algorithms for Rank-1 Bimatrix Games
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The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. For a game of rank k, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank k-1 with a hyperplane. We comprehensively analyze games of rank one. They are economonically more interesting than zero-sum games (which have rank zero), but are nearly as easy to solve. One equilibrium of a rank-1 game can be found in polynomial time. All equilibria of a rank-1 game can be found by path-following, which finds only one equilibrium of a bimatrix game. The number of equilibria of a rank-1 game may be exponential, but is polynomial in expectation when payoffs are slightly perturbed. We also present a new rank-preserving homeomorphism between bimatrix games and their equilibrium correspondence.
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