Duong Manh Hong, Tran Hoang Minh, Han The Anh
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK.
Data Analytics Department, Esmart Systems, 1783, Halden, Norway.
J Math Biol. 2019 Jan;78(1-2):331-371. doi: 10.1007/s00285-018-1276-0. Epub 2018 Aug 1.
The analysis of equilibrium points is of great importance in evolutionary game theory with numerous practical ramifications in ecology, population genetics, social sciences, economics and computer science. In contrast to previous analytical approaches which primarily focus on computing the expected number of internal equilibria, in this paper we study the distribution of the number of internal equilibria in a multi-player two-strategy random evolutionary game. We derive for the first time a closed formula for the probability that the game has a certain number of internal equilibria, for both normal and uniform distributions of the game payoff entries. In addition, using Descartes' rule of signs and combinatorial methods, we provide several universal upper and lower bound estimates for this probability, which are independent of the underlying payoff distribution. We also compare our analytical results with those obtained from extensive numerical simulations. Many results of this paper are applicable to a wider class of random polynomials that are not necessarily from evolutionary games.
平衡点的分析在进化博弈论中非常重要，在生态学、群体遗传学、社会科学、经济学和计算机科学等领域有着众多实际应用。与以往主要侧重于计算内部平衡点预期数量的分析方法不同，本文研究了多人双策略随机进化博弈中内部平衡点数量的分布。我们首次推导出了一个封闭公式，用于计算博弈具有特定数量内部平衡点的概率，该公式适用于博弈收益项的正态分布和均匀分布。此外，利用笛卡尔符号法则和组合方法，我们给出了该概率的几个通用上下界估计，这些估计与潜在的收益分布无关。我们还将分析结果与通过广泛数值模拟得到的结果进行了比较。本文的许多结果适用于更广泛的一类随机多项式，这些多项式不一定来自进化博弈。
