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 payo matrix behave either as uniform,
or normally distributed 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 a stability property
that the payo submatrix indicated by the support of an ESS must satisfy.
We then combine this result with a recent result of McLennan and Berg (2005), concerning the
expected number of Nash Equilibria in normalrandom bimatrix games, to show that the expected
number of ESS is signicantly smaller than the expected number of symmetric Nash equilibria of the
underlying symmetric bimatrix game.