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. We illustrate our theory using a variety of case studies, including a novel, continuous variant of the war of attrition game.

翻译：本文研究玩家拥有连续策略集的群体博弈演化动力学。在该设定中，模型对应于概率测度流形上的无穷维微分方程。我们将演化博弈的耗散性理论从有限策略集推广至紧度量空间上的无限策略集，并推导出无穷维动力学下纳什均衡稳定性的充分条件。该分析适用于广泛类别的演化博弈，且具有模块化特征——动力学与博弈收益结构的相关条件可独立验证。通过将理论特化至单调博弈类，我们重新推导了Brown-von Neumann-Nash动力学与无偏成对比较动力学的现有稳定性结论。此外，我们将理论拓展至动态收益模型，进一步扩展了框架的适用性。通过包含新型连续型消耗战博弈在内的多案例研究，我们对理论进行了验证。