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 ran-

domly 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, nor-

mal, exponential); and (ii) it converges to 1 − 1/√e for distributions

F with “slower than exponential decreasing tails” (e.g., lognormal,

Pareto, Cauchy).