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We consider evolutionary dynamics for population games in which players havea continuum of strategies at their disposal. Models in this setting amount toinfinite-dimensional differential equations evolving on the manifold ofprobability measures. We generalize dissipativity theory for evolutionary gamesfrom finite to infinite strategy sets that are compact metric spaces, andderive sufficient conditions for the stability of Nash equilibria under theinfinite-dimensional dynamics. The resulting analysis is applicable to a broadclass of evolutionary games, and is modular in the sense that the pertinentconditions on the dynamics and the game's payoff structure can be verifiedindependently. By specializing our theory to the class of monotone games, werecover as special cases existing stability results for the Brown-vonNeumann-Nash and impartial pairwise comparison dynamics. We also extend ourtheory to models with dynamic payoffs, further broadening the applicability ofour framework. We illustrate our theory using a variety of case studies,including a novel, continuous variant of the war of attrition game.

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