A game-theoretic and dynamical-systems analysis of selection methods in coevolution

We use evolutionary game theory (EGT) to investigate the dynamics and equilibria of selection methods in coevolutionary algorithms. The canonical selection method used in EGT is equivalent to the standard "fitness-proportional" selection method used in evolutionary algorithms. All attractors of the EGT dynamic are Nash equilibria; we focus on simple symmetric variable-sum games that have polymorphic Nash-equilibrium attractors. Against the dynamics of proportional selection, we contrast the behaviors of truncation selection, (μ,λ),(μ+λ), linear ranking, Boltzmann, and tournament selection. Except for Boltzmann selection, each of the methods we test unconditionally fail to achieve polymorphic Nash equilibrium. Instead, we find point attractors that lack game-theoretic justification, cyclic dynamics, or chaos. Boltzmann selection converges onto polymorphic Nash equilibrium only when selection pressure is sufficiently low; otherwise, we obtain attracting limit-cycles or chaos. Coevolutionary algorithms are often used to search for solutions (e.g., Nash equilibria) of games of strategy; our results show that many selection methods are inappropriate for finding polymorphic Nash solutions to variable-sum games. Another application of coevolution is to model other systems; our results emphasize the degree to which the model\´s behavior is sensitive to implementation details regarding selection-details that we might not otherwise believe to be critical.