Evolutionarily stable strategies of random games, and the vertices of random polygons

An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative ("mutant") strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for "almost every large" game? Letting the entries in the n × n game matrix be independently randomly chosen according to a distribution F, we study the number of ESS with support of size 2. In particular, we show that, as n → ∞, the probability of having such an ESS: (i) converges to 1 for distributions F with "exponential and faster decreasing tails" (e.g., uniform, normal, exponential); and (ii) converges to 1 - 1/√e for distributions F with "slower than exponential decreasing tails" (e.g., lognormal, Pareto, Cauchy). Our results also imply that the expected number of vertices of the convex hull of n random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).