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Evolutionarily Stable Strategies of Random Games, and the Vertices of Random Polygons

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  • Sergiu Hart
  • Yosef Rinott
  • Benjamin Weiss

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 x n game matrix be randomly chosen according to an underlying distribution F, we study the number of ESS with support of size 2. In particular, we show that, as n goes to infinity, 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) it 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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File URL: http://www.ma.huji.ac.il/hart/papers/ess.pdf
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Paper provided by UCLA Department of Economics in its series Levine's Bibliography with number 321307000000000781.

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Date of creation: 26 Jan 2007
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Handle: RePEc:cla:levrem:321307000000000781

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  1. Mark Bagnoli & Ted Bergstrom, 2005. "Log-concave probability and its applications," Economic Theory, Springer, vol. 26(2), pages 445-469, 08.
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