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

• 成果类型: Article
• 署名作者: Hart, Sergiu; Rinott, Yosef; Weiss, Benjamin
• 署名单位: Hebrew University of Jerusalem; Hebrew University of Jerusalem; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP455
• 发表日期: 2008
• 页码: 259-287
• 关键词: convex hulls; points

摘要

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 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 --> 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) converges to 1 - 1 1/root 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).