Limit Theory for $N$-Player $\alpha$-Potential Games

Abstract

The recently introduced framework of $\alpha$-potential games facilitates the analysis of finite-player dynamic games by reducing the search for approximate Nash equilibria to the minimization of a single $\alpha$-potential function. In this work, we investigate the large population limit of $\alpha$-potential games, and show that potential mean field games (MFGs) arise naturally. Specifically, we show that both the optimal values and the minimizers of normalized $N$-player $\alpha_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We further show that $\lim_{N\to\infty}\alpha_N= 0$ is equivalent to standard conditions for potential MFGs, and provide a unified construction of potential functions for MFGs. A key technical ingredient is the establishment of a Poincar\'e lemma for Wasserstein space. We also establish that the objective of the limiting MFC problem is a potential function for the corresponding MFGs. Together, our results not only yield new constructions of potential MFGs from finite-player games through the asymptotic condition $\lim_{N\to \infty}\alpha_N= 0$, but also establish propagation of chaos from $N$-player games to MFGs for general controlled diffusions with common noise and non-separable control interactions.