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

TL;DR

This paper investigates the existence of evolutionarily stable strategies (ESS) in large random games, showing that for most distributions, ESS with support size 2 become almost certain as game size grows, with implications for convex hull vertices.

Contribution

It provides asymptotic probabilities for the existence of ESS in large random games based on the tail behavior of the distribution of game entries.

Findings

Probability of ESS approaches 1 for exponential or faster tail distributions.

Probability of ESS approaches 1 - 1/√e for slower tail distributions.

Expected number of convex hull vertices converges to infinity or 4 depending on distribution tail behavior.

Abstract

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).