Deterministic and Stochastic Equilibrium Selection in Evolutionary Game Theory

该奖项资助了在进化博弈论中追求均衡选择的两个新方向的研究。本课题第一部分介绍了确定性动力学下的均衡选择模型，第二部分考虑了随机稳定性下的均衡选择。PI和他的合著者首先介绍了一类新的确定性进化博弈动力学，称为抽样最佳反应动力学。为了定义它们，他们假设当一个代理考虑切换动作时，他观察的是固定数量或随机抽样的对手的动作。他将样本中行为的经验分布视为对总体中行为分布的估计，并选择一种最优的行为来对抗这种经验分布。该团队表明，在某些具有多个严格均衡的博弈中，存在一个均衡几乎是全局渐近稳定的，它吸引了所有内部初始条件的解。由于分析是确定性的，他们得到的预测需要很少的时间才能变得相关。该团队还调查了确定性选择结果可以从其他修订规则中获得的程度。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。在具有突变的最佳对策模型中，次优选择的概率独立于其收益结果，可以使用突变计数参数进行随机稳定性分析。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骤的数量和每一步的不可能性。正因为如此，除了双策略博弈，我们对均衡选择知之甚少。研究人员认为，通过在双重限制下研究随机稳定性，同时在代理中具有噪音水平？当决策变小而种群规模变大时，可以结合大偏差理论和最优控制理论的技术来评估均衡之间过渡的概率，从而确定随机稳定状态。他们还认为，平稳分布的渐近性质，以及由此产生的随机稳定状态的同一性，与所选极限的阶数无关。这种分析可以将现有的双策略游戏的简单结果扩展到具有任意数量策略的游戏。更广泛的影响在具有大量相互作用主体的环境中，包括具有多边外部性和宏观经济背景的环境，多重均衡的存在可能导致效率低下和无法预测行为。通过发展决策的动态模型，在这些环境中得出独特的预测，个人指数提供了工具，可以帮助规划者通过精心设计激励和信息提供政策来实现社会目标。这项研究包括为博弈论的研究和教学开发技术和教学材料的组成部分：一套易于使用的开源软件，用于构建与进化博弈动力学相关的阶段图和其他图形。该提案的这一部分使进化博弈技术更容易被经济学、生物学、工程学和其他领域的理论和应用工作者所接受，项目的持续发展将进一步扩大其效用和范围。

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.