Deterministic and Stochastic Equilibrium Selection in Evolutionary Game Theory

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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.