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 independent random variables, almost all ESS have support sizes o(n), where n is the number of possible types for a player. Our arguments are based exclusively on the severity of a stability property that the payoff submatrix indicated by the support of an ESS must satisfy. We then combine our normal–random result with a recent result of McLennan and Berg (2005), 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. JEL Classification Code: C7 – Game Theory and Bargaining Theory.