Finding a Nash equilibrium of a random win-lose game in expected polynomial time

A long-standing open problem in algorithmic game theory asks whether or not there is a polynomial time algorithm to compute a Nash equilibrium in a random bimatrix game. We study random win-lose games, where the entries of the $n\times n$ payoff matrices are independent and identically distributed (i.i.d.) Bernoulli random variables with parameter $p=p(n)$. We prove that, for nearly all values of the parameter $p=p(n)$, there is an expected polynomial-time algorithm to find a Nash equilibrium in a random win-lose game. More precisely, if $p\sim cn^{-a}$ for some parameters $a,c\ge 0$, then there is an expected polynomial-time algorithm whenever $a\not\in \{1/2, 1\}$. In addition, if $a = 1/2$ there is an efficient algorithm if either $c \le e^{-52} 2^{-8} $ or $c\ge 0.977$. If $a=1$, then there is an expected polynomial-time algorithm if either $c\le 0.3849$ or $c\ge \log^9 n$.