随机对策的进化稳定策略和随机多边形的顶点
Evolutionarily stable strategies of random games, and the vertices of random polygons
摘要：
进化稳定策略（ESS）是一种不受稀有替代策略（“mux”）入侵的均衡策略。与纳什均衡不同，ESS并不总是存在于有限对策中。在本文中，我们解决的问题，会发生什么事情时，游戏的大小增加：是否存在ESS的“几乎每一个大的”游戏？让$n\times n$游戏矩阵中的条目根据分布$F$独立随机选择，我们研究了支持$2.$大小的ESS的数量。特别是，我们表明，作为$n\to \infty$，具有这样的ESS的概率：（i）对于具有“指数且更快递减尾部”的分布$F$收敛于1（例如，均匀、正态、指数）;以及（ii）对于具有"慢于指数递减尾部“的分布$F$收敛到$1-1/\sqrt{e}$（例如，对数正态分布，帕累托，柯西）。我们的结果还意味着，在平面上的$n$随机点的凸包的期望顶点数收敛到无穷大的分布（i），和4的分布（ii）。
英文摘要：
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 $n\times n$ 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).
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