随机空间演化对策的确定性方程

On the expected number of real roots of random polynomials arising from evolutionary game theory

In this paper, we obtain finite estimates and asymptotic formulas for the expected number of real roots of two classes of random polynomials arising from evolutionary game theory. As a consequence of our analysis, we achieve an asymptotic formula for the expected number of internal equilibria in multi-player two-strategy random evolutionary games. Our results contribute both to evolutionary game theory and random polynomial theory.

随机演化博弈与随机多项式

Random evolutionary games and random polynomials

本文发现随机非对称演化博弈的均衡多项式恰为 Kostlan-Shub-Smale 系统，并借此解析刻画内部均衡数量的统计规律，证明对称性会增加内部均衡期望数量。

随机演化博弈中共存子空间上的扩散动力学

Difffusion Dynamics on the Coexistence Subspace in a Stochastic Evolutionary Game

频率依赖性选择反映了不同物种在争夺环境中有限资源时的相互作用。在随机进化博弈中，物种的相对适应度引导着由于随机漂移而波动的进化动力学。物种选择优势对环境的依赖为进化动力学引入了额外的可能性。我们分析了一个简单的模型，其中一个随机的环境允许竞争物种共存很长一段时间之前，固定的一个单一的物种发生。在我们的分析中，我们使用的竞争物种的线性组合的稳定性近似的随机动力系统的扩散的一维共存区域。我们的方法显着简化了计算的概率第一灭绝和预期的时间，并展示了严格的模型降阶技术，用于评估准平稳随机演化模型的属性。

The replicator equation in stochastic spatial evolutionary games

We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size $N\to \infty$. The model is a voter model perturbation. For typical populations, we require perturbation strengths satisfying $1/N\ll w\ll 1$. Under appropriate conditions on the space, the limiting density processes of strategy are proven to obey the replicator equation, and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the bi…

Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

In this paper, we study analytically the statistics of the number of equilibria in pairwise social dilemma evolutionary games with mutation where a game's payoff entries are random variables. Using the replicator-mutator equations, we provide explicit formulas for the probability distributions of the number of equilibria as well as other statistical quantities. This analysis is highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g., cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff entries is difficult. Our d…

Persistence probability of a random polynomial arising from evolutionary game theory

In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.