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
