Rank-1 bimatrix games

Abstract

Given a rank-1 bimatrix game (A, B), i.e., where rank(A + B) = 1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in [8,19]. In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. Our approach also provides new proofs of important classical results such as the existence and oddness of Nash equilibria, and the index theorem for bimatrix games. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0, 1] k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

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“…To relate to other conditions considered in the literature, we point out here that perturbation stability is incomparable to the class of constant-rank games studied, e. g., in [25,1]. One can generate n-by-n matrices R and C that are highly stable and yet where R +C has large rank (rank n), and one can also generate zero-sum games that are very unstable under perturbations.…”

“…Their result implies that in random two-player games, Nash equilibria can be computed in expected polynomial time. Kannan and Theobald [25] provide an FPTAS for the case where the sum of the two payoff matrices has constant rank and Adsul et al [1] provide a polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. Note that it is possible for a game to be stable and yet have high rank, and on the other hand to be unstable and have rank 0; see Appendix B.…”

“…, v (k 1 ) such thatp · v ≥p · v + ε is at most 1/3. Consider an arbitrary distribution p in {p (1) , . .…”

“…To see this consider adding ε to R [1,0]. That is, if the Row player plays action 1 and the Column player plays action 0, then the payoff to Row player is −0.25/n + ε > 0.…”

“…The starting point of our work is the realization that games are typically only abstractions of reality and except in the most controlled settings, payoffs listed in a game that represents an interaction between self-interested agents are only approximations to the agents' exact utilities. 1 As a result, for problems such as equilibrium computation, it is natural to focus attention on games that are robust to the exact payoff values, in the sense that small changes to the entries in the game matrices do not cause the Nash equilibria to fluctuate wildly; otherwise, even if equilibria can be computed, they may not actually be meaningful for understanding behavior in the game that is played. In this work, we focus on such games and analyze their structural properties as well as their implications to the equilibrium computation problem.…”