Rationality, Irrationality, and Transition Dynamics in Evolutionary Game Theory

该项目由进化博弈论的四项研究组成，进化博弈论是一个对参与反复战略互动的大量代理人的行为进行建模的领域。前三项研究被设定在一个统一的大人口建模框架中。种群博弈由有限的动作集和每个动作的收益函数定义；收益连续地取决于代理人的动作选择的分布。为了避免在这个大人口背景下的均衡博弈假设，假设每个代理人偶尔都有机会修改他的策略选择。在这样的时刻，使用修订协议来描述代理对新策略的选择，该修订协议指定了每对策略之间的切换的条件速率。在适度的时间跨度内，聚合行为可以很好地用一个确定性的微分方程--平均动态方程来近似，这个方程是由群体行为的预期变化定义的。但对很长时间跨度的行为分析必须明确说明修订过程中固有的随机性。消除严格支配的策略是标准博弈论分析中采用的最温和的要求。令人惊讶的是，研究1认为，几乎没有确定性的进化动力学消除了严格支配的策略。结果表明，在某些博弈中，任何满足三个自然条件--连续性、正相关性和创新性--的动态都不能消除严格支配策略。显式构造了占优策略生存的博弈，并说明了为什么已有的正向结果对选择规则的微小变化不是稳健的。研究2研究了n个策略势博弈的Logit演化。与通常的突变模型不同，Logit模型中错误选择的概率取决于相应的收益损失。这一假设导致了更现实但更复杂的不平衡动力学；它使均衡之间的转变分析复杂化，从而使均衡选择的分析复杂化。利用大偏差理论的技巧，从稳定均衡的吸引盆地刻画了逃脱的概率和等待时间。这使得能够精确地表征极限平稳分布，从而能够精确地表征随机稳定状态。研究3解决了进化博弈论中的一个基本问题：纳什均衡是否能与近视、部分信息和不完全反应的大群体中的静态行为相一致。本文构造了一类演化动力学模型，其休止点正是潜在博弈的纳什均衡。这种动态是基于持续修订的协议，只需要代理知道他们当前和未来策略的回报。研究4应用进化博弈论中的技术来研究居住隔离的动态。在一篇开创性的论文中，谢林(1971)展示了即使代理人只有轻微的偏好与他们自己的群体成员住在一起，种族隔离也会出现。研究人员最近开发了用于贝叶斯博弈进化分析的工具，这些工具被用来重新制定和扩展谢林的工作，允许内生确定外部社区的特征，以及对社区组成的更广泛的偏好。该提案最后提出了一个组件，为博弈论的研究和教学开发技术和教学材料：即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。

This project consists of four studies in evolutionary game theory, a field that models the behavior of large populations of agents engaged in recurring strategic interactions. The first three studies are set in a unified framework for large population modeling. A population game is defined by a finite set of actions and a payoff function for each; payoffs depend continuously on the distribution of agents' action choices. To avoid the assumption of equilibrium play in this large population context, it is assumed that each agent occasionally receives an opportunity to revise his choice of strategy. At such moments, the agent's selection of a new strategy is described using a revision protocol, which specifies the conditional rates of switches between each pair of strategies. Over moderate time spans, aggregate behavior is well approximated by a deterministic differential equation, the mean dynamic, which is defined by the expected changes in the population's behavior. But analyses of behavior over very long time spans must account explicitly for the randomness inherent in the revision process.Elimination of strictly dominated strategies is the mildest requirement employed in standard game-theoretic analysis. Surprisingly, Study 1 argues that few deterministic evolutionary dynamics eliminate strictly dominated strategies. It is shown that any dynamic satisfying three natural conditions--continuity, positive correlation, and innovation--fails to eliminate strictly dominated strategies in some games. Games are explicitly constructed in which dominated strategies survive, and it is shown why existing positive results are not robust to small changes in choice rules.Study 2 studies logit evolution in n strategy potential games. Unlike in the usual mutation model, probabilities of errant choices in the logit model depend on the corresponding payoff losses. This assumption leads to more realistic but more complex disequilibrium dynamics; it complicates the analysis of transitions between equilibria and, consequently, the analysis of equilibrium selection. Using techniques from large deviations theory, the probabilities of and waiting times before escapes are characterized from the basins of attraction of stable equilibria. This enables a precise characterization of the limiting stationary distribution, and thus of the stochastically stable state. Geometric methods are used to determine the rate of convergence of the evolutionary process to its limit distribution.Study 3 addresses a basic question in evolutionary game theory: whether Nash equilibrium can identified with stationary behavior in large populations of myopic, partially informed, and imperfectly responsive agents. In this study, a class of evolutionary dynamics is constructed, whose rest points are precisely the Nash equilibria of the underlying game. The dynamics are based on continuous revision protocols that only require agents to know the payoffs of their current and prospective strategies. Study 4 applies techniques from evolutionary game theory to investigate the dynamics of residential segregation. In a seminal paper, Schelling (1971) showed how segregation can arise even when agents have only a slight preference for residing with members of their own group. The researchers have recently developed tools for the evolutionary analysis of Bayesian games, and they are used to reformulate and extend Schelling's work, allowing both for endogenous determination of the characteristics of outside neighborhoods and for a wider array of preferences over neighborhood compositions.The proposal concludes with a component that develops technology and instructional materials for research and teaching in game theory: namely, a suite of easy-to-use, open source software for constructing phase diagrams and other graphics related to evolutionary game dynamics.