NP-hardness of Computing Uniform Nash Equilibria on Planar Bimatrix Games

We study the complexity of computing a uniform Nash equilibrium on a bimatrix game. It is known that such a problem is NP-complete even if a bimatrix game is win-lose [BIL08]. Fortunately, if a win-lose bimatrix game is planar, then uniform Nash equilibria always exist. We have a polynomial-time algorithm for finding a uniform Nash equilibrium of a planar win-lose bimatrix game [AOV07]. The following question is left: How hard to determine the existence of uniform Nash equilibria of a planar bimatrix game not necessarily win-lose? This paper resolves this issue. We prove that the problem of deciding whether a planar bimatrix game such that both players' payoff matrices consist of two types of non-zero elements has uniform Nash equilibria is also NP-complete.