Settling Some Open Problems on 2-Player Symmetric Nash Equilibria

TL;DR

This paper proves that finding and counting non-symmetric Nash equilibria in symmetric games are computationally hard, and it resolves the open problem of computing symmetric rank 1 Nash equilibria in polynomial time.

Contribution

It establishes NP-completeness for finding non-symmetric NE and #P-completeness for counting them, and provides a polynomial-time algorithm for symmetric rank 1 games.

Findings

Finding non-symmetric NE in symmetric games is NP-complete.

Counting non-symmetric NE in symmetric games is #P-complete.

Polynomial-time algorithm for symmetric rank 1 games.

Abstract

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 #P-complete. 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 solve the case of symmetric rank 1 games. We resolve this problem.