Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49-73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately (πlogN/2)M−1/M. Letting M=N→∞, the expected number of Nash equilibria is exp(NM+O(logN)), where M≈0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.