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

Sergiu Hart, Yosef Rinott and Benjamin Weiss

Papers from arXiv.org

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).

Date: 2008-01
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Citations: View citations in EconPapers (3)

Published in Annals of Applied Probability 2008, Vol. 18, No. 1, 259-287

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Related works:
Working Paper: Evolutionarily Stable Strategies of Random Games, and the Vertices of Random Polygons (2007)
Working Paper: Evolutionarily Stable Strategies of Random Games, and the Vertices of Random Polygons (2007)
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