Limit Theory for $N$-Player $α$-Potential Games
The recently introduced framework of $α$-potential games facilitates the analysis of finite-player dynamic games by reducing the search for approximate Nash equilibria to the minimization of a single $α$-potential function. In this work, we show that the approximate potential structure of finite-player games persists in the large-population limit, leading naturally to a notion of $α$-potential mean field games (MFGs). Specifically, we show that both the optimal values and the minimizers of normalized $N$-player $α_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We identify the objective of this limiting MFC problem as an $α_\infty$-potential function for an MFG, and characterize the parameter $α_\infty$ analytically. We further show that $\lim_{N\to\infty}α_N= 0$ is equivalent to standard conditions for potential MFGs, in which case the limiting MFC objective becomes a potential function for the MFG. Together, these results provide new constructions of potential MFGs from finite-player games through the asymptotic condition $\lim_{N\to \infty}α_N= 0$, and establish propagation of chaos of potential-selected equilibria from $N$-player games to MFGs with general controlled diffusions, common noise, and non-separable state-control interactions.