A finite, two-player, nondegenerate, symmetric game is defined by a nondegenerate $n \times n$ *payoff matrix* $A$. If player 1 plays strategy $i$ and player 2 plays strategy $j$, then player 1's payoff is $A_{ij}$ and player 2's payoff is $A_{ji}$. It is well known that the problem of computing a symmetric Nash Equillibrium in such a game is PPAD-complete (PPAD lies between P and NP but is probably intractable).

Wilson's *Oddness Theorem* states that there are an odd number of symmetric Nash Equilibria in such games. This gives rise to my question. Suppose we have found two equilibria of $A$. Given these, what is the computational complexity of computing one more?

Or, more generally - given $2k$ equilibria, what is the complexity of computing another?