Evolutionarily stable strategies of random games, and the vertices of random polygons
Sergiu Hart, Yosef Rinott, and Benjamin Weiss
Source: Ann. Appl. Probab.
Volume 18, Number 1
(2008), 259-287.
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×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 
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).
Primary Subjects: 91A22, 60D05
Secondary Subjects: 60F99, 52A22
Keywords: Evolutionarily stable strategy; ESS; random game; random polytope; convex hull of random points; Nash equilibrium; Poisson approximation; Chen–Stein method; heavy-tailed distribution; subexponential distribution; threshold phenomenon
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1199890023
Digital Object Identifier: doi:10.1214/07-AAP455
Mathematical Reviews number (MathSciNet):
MR2380899
Zentralblatt MATH identifier:
1132.91352
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