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.
CITATION STYLE
Mehta, R., Vazirani, V. V., & Yazdanbod, S. (2015). Settling some open problems on 2-player symmetric nash equilibria. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9347, pp. 272–284). Springer Verlag. https://doi.org/10.1007/978-3-662-48433-3_21
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