Evolutionary Games on Infinite Strategy Sets: Convergence to Nash Equilibria via Dissipativity

Abstract

We consider evolutionary dynamics for population games in which players have a continuum of strategies at their disposal. Models in this setting amount to infinite-dimensional differential equations evolving on the manifold of probability measures. We generalize dissipativity theory for evolutionary games from finite to infinite strategy sets that are compact metric spaces, and derive sufficient conditions for the stability of Nash equilibria under the infinite-dimensional dynamics. The resulting analysis is applicable to a broad class of evolutionary games, and is modular in the sense that the pertinent conditions on the dynamics and the game's payoff structure can be verified independently. By specializing our theory to the class of monotone games, we recover as special cases existing stability results for the Brown-von Neumann-Nash and impartial pairwise comparison dynamics. We also extend our theory to models with dynamic payoffs, further broadening the applicability of our framework. Throughout our analyses, we identify and elaborate on new technical conditions that are key in extending dissipativity theory from finite to infinite strategy sets, such as compactness of the set of Nash equilibria and evolution of dynamic payoffs within a compact positively invariant set. We illustrate our theory using a variety of case studies, including a novel, continuous variant of the war of attrition game.