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


  • Sergiu Hart
  • Yosef Rinott
  • Benjamin Weiss


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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  • Sergiu Hart & Yosef Rinott & Benjamin Weiss, 2007. "Evolutionarily Stable Strategies of Random Games, and the Vertices of Random Polygons," Levine's Bibliography 321307000000000781, UCLA Department of Economics.
  • Handle: RePEc:cla:levrem:321307000000000781

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    1. Mark Bagnoli & Ted Bergstrom, 2005. "Log-concave probability and its applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 445-469, August.
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    Cited by:

    1. 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.
    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.

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