Evolutionarily Stable Strategies of Random Games, and the Vertices of Random Polygons
AbstractAn 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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Bibliographic InfoPaper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp445.
Date of creation: Jan 2007
Date of revision:
Publication status: Published in Annals of Applied Probability 18 (2008), 1, 259-287
Other versions of this item:
- 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.
- NEP-ALL-2007-02-17 (All new papers)
- NEP-EVO-2007-02-17 (Evolutionary Economics)
- NEP-GTH-2007-02-17 (Game Theory)
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- Mark Bagnoli & Ted Bergstrom, 2005.
"Log-concave probability and its applications,"
Springer, vol. 26(2), pages 445-469, 08.
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