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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 $n\times n$ 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).

Suggested Citation

  • Sergiu Hart & Yosef Rinott & Benjamin Weiss, 2008. "Evolutionarily stable strategies of random games, and the vertices of random polygons," Papers 0801.3353, arXiv.org.
  • Handle: RePEc:arx:papers:0801.3353
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    References listed on IDEAS

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    1. Mark Bagnoli & Ted Bergstrom, 2006. "Log-concave probability and its applications," Studies in Economic Theory, in: Charalambos D. Aliprantis & Rosa L. Matzkin & Daniel L. McFadden & James C. Moore & Nicholas C. Yann (ed.), Rationality and Equilibrium, pages 217-241, Springer.
    2. Devroye, Luc, 1991. "On the oscillation of the expected number of extreme points of a random set," Statistics & Probability Letters, Elsevier, vol. 11(4), pages 281-286, April.
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    Cited by:

    1. Sam Ganzfried, 2018. "Optimization-Based Algorithm for Evolutionarily Stable Strategies against Pure Mutations," Papers 1803.00607, arXiv.org, revised Jan 2019.
    2. Manh Hong Duong & The Anh Han, 2016. "On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game," Dynamic Games and Applications, Springer, vol. 6(3), pages 324-346, September.
    3. Ohad Navon, 2016. "Evolutionarily Stable Strategies of Random Games and the Facets of Random Polytopes," Discussion Paper Series dp702, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

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