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 R I, the expected number of Nash equilibria is approximately (R(π log N) / 2)(M-1)/R M. Letting M = N R I, the expected number of Nash equilibria is exp(NM + O(log N)), where M A 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies. © 2004 Elsevier Inc. All rights reserved.

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