Settling some open problems on 2-player symmetric nash equilibria

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and the problem of counting the number of non-symmetric NE in a symmetric game is #Pcomplete. In 2005, Kannan and Theobald defined the rank of a bimatrix game represented by matrices (A,B) to be rank(A + B) and asked whether a NE can be computed in rank 1 games in polynomial time. Observe that the rank 0 case is precisely the zero sum case, for which a polynomial time algorithm follows from von Neumann’s reduction of such games to linear programming. In 2011, Adsul et al. obtained an algorithm for rank 1 games; however, it does not guarantee symmetric NE in symmetric rank 1 game. We resolve this problem.