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

Sergiu Hart, Yosef Rinott, Benjamin Weiss

Arxiv ID: 0801.3353•Last updated: 9/22/2022

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 $n\times n$
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 $n\to \infty$, 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 $1-1/\sqrt{e}$ 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).

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