Deterministic and Stochastic Equilibrium Selection in Evolutionary Game Theory

该奖项资助了为进化博弈论中的均衡选择寻找两个新方向的研究。该项目的第一部分介绍了确定性动态下的均衡选择模型，第二部分考虑了随机稳定下的均衡选择，Pi和他的合著者首先引入了一类新的确定性进化博弈动力学，称为抽样最佳响应动力学。为了定义它们，他们假设当代理人考虑切换动作时，他观察到固定或随机抽样的对手的动作。他将样本中行动的经验分布视为对总体中行动分布的估计，并根据这种经验分布选择最优的行动。该团队证明，在具有多个严格均衡的某些博弈中，存在一个几乎全局渐近稳定的均衡，吸引来自所有内部初始条件的解。由于分析是确定性的，他们获得的预测只需很短的时间就能变得相关。该团队还调查了从其他修订规则获得的动力学确定性选择结果的程度。项目的第二部分考虑了基于噪声最佳响应规则的进化模型的随机稳定性。在带有突变的最佳反应模型中，次优选择的概率与其收益结果无关，随机稳定性分析可以使用突变计数参数来进行。但是，当次优选择的概率取决于其收益结果时，在均衡之间遵循给定路径的概率取决于步数和每一步的不可能性。正因为如此，除了两策略博弈之外，人们对均衡选择知之甚少。研究人员认为，通过研究双重极限下的随机稳定性，既有代理人中的噪声水平？当决策变小，种群规模变大时，可以结合大偏差理论和最优控制理论的技术来估计平衡点之间转换的概率，从而确定随机稳定状态。他们还认为，平稳分布的渐近性质，以及随机稳定状态的一致性，与所选择的极限顺序无关。这种分析将把两策略博弈的更简单的现有结果推广到具有任意策略数量的博弈。广泛影响在具有大量相互作用的主体的环境中，包括具有多边外部性和宏观经济背景的环境中，多个均衡的存在可能导致效率低下并且无法预测行为。通过开发动态决策模型，在这些环境中产生独特的预测，PI提供了工具，可以帮助规划者通过仔细制定激励和信息提供政策来实现社会目标。这项研究包括一个为博弈论的研究和教学开发技术和教学材料的组件：一套易于使用的开源软件，用于构建相图和其他与进化博弈动力学相关的图形。该提案的这一组成部分使进化博弈技术更容易为经济学、生物学、工程学和其他领域的理论和应用工作者所接受，该项目的持续发展将进一步扩大其实用性和范围。

This award funds research pursuing two new directions for equilibrium selection in evolutionary game theory. The first part of the project introduces models of equilibrium selection under deterministic dynamics, and the second considers equilibrium selection via stochastic stability.The PI and his co-authors first introduce a new class of deterministic evolutionary game dynamics called sampling best response dynamics. To define them, they assume that when an agent considers switching actions, he observes the actions of a fixed or random number of randomly sampled opponents. He views the empirical distribution of actions in his sample as an estimate of the distribution of actions in the population, and chooses an action that is optimal against this empirical distribution. The team shows that in certain games with multiple strict equilibria, there is one equilibrium that is almost globally asymptotically stable, attracting solutions from all interior initial conditions. Since the analysis is deterministic, the predictions they obtain require little time to pass to become relevant. The team also investigates the extent to which deterministic selection results can be obtained for dynamics derived from other revision rules.The second part of the project considers stochastic stability in models of evolution based on noisy best response rules. In models of best responses with mutations, in which the probability of a suboptimal choice is independent of its payoff consequences, stochastic stability analysis can proceed using mutation counting arguments. But when the probability of a suboptimal choice depends on its payoff consequences, the probability of following a given path between equilibria depends on both the number of steps and the unlikelihood of each step. Because of this, little is known about equilibrium selection beyond two-strategy games. The researchers argue that by studying stochastic stability in double limits, having both the level of noise in agents? decisions become small and the population size become large, one can combine techniques from large deviations theory and optimal control theory to evaluate the probabilities of transitions between equilibria, and so determine the stochastically stable states. They also argue that the asymptotic properties of the stationary distribution, and hence the identity of the stochastically stable states, is independent of the order of limits chosen. This analysis would extend simpler existing results for two-strategy games to games with arbitrary numbers of strategies.Broader ImpactsIn environments with large numbers of interacting agents, including settings with multilateral externalities and macroeconomic contexts, the existence of multiple equilibria can lead to inefficiency and to an inability to predict behavior. By developing dynamic models of decision that lead to unique predictions in these settings, the PI provides tools that could help planners attain social goals through the careful crafting of incentives and information-provision policies. The research includes a component that develops technology and instructional materials for research and teaching in game theory: a suite of easy-to-use, open source software for constructing phase diagrams and other graphics related to evolutionary game dynamics. Thiscomponent of the proposal has made evolutionary game techniques more accessible to theoretical and applied workers in economics, biology, engineering, and other fields, and continued development of the project will further its utility and scope.