On the support size of stable strategies in random games

In this paper we study the support sizes of evolutionary stable strategies (ESS) in random evolutionary games. We prove that, when the elements of the payoff matrix behave either as uniform, or normally distributed random variables, almost all ESS have support sizes , where n is the number of possible types for a player. Our arguments are based exclusively on a stability property that the payoff submatrix indicated by the support of an ESS must satisfy.We then combine this result with a recent result of McLennan and Berg [A. McLennan, J. Berg, The asymptotic expected number of nash equilibria of two player normal form games, Games and Economic Behavior 51 (2005) 264–295], concerning the expected number of Nash Equilibria in normal-random bimatrix games, to show that the expected number of ESS is significantly smaller than the expected number of symmetric Nash equilibria of the underlying symmetric bimatrix game.