An Overview of Some Extensions of Mean Field Games beyond Perfect Homogeneity and Anonymity
Mathieu Laurière
TL;DR
This work surveys extensions of the mean field games framework beyond homogeneous and anonymous populations, including multi-population MFGs, graphon MFGs, major-minor and Stackelberg formulations, and cooperative variants. It presents both finite-player and asymptotic models, highlights fixed-point and forward–backward structures (e.g., Riccati and coupled ODEs) used to characterize equilibria, and discusses numerical approaches and key theoretical challenges. The contributions clarify how non-exchangeable interactions and influential actors alter equilibrium concepts and mean-field limits, and they connect MFG with mean field control and mean field type games to capture social optima and cooperative settings. The practical impact lies in more realistic modeling of heterogeneous, networked populations in domains such as traffic, crowds, economics, and epidemic management, where heterogeneity and non-anonymity are essential.
Abstract
The mean field games (MFG) paradigm was introduced to provide tractable approximations of games involving very large populations. The theory typically rests on two key assumptions: homogeneity, meaning that all players share the same dynamics and cost functions, and anonymity, meaning that each player interacts with others only through their empirical distribution. While these assumptions simplify the analysis, they can be restrictive for many applications. Fortunately, several extensions of the standard MFG framework that relax these assumptions have been developed in the literature. The purpose of these notes is to offer a pedagogical introduction to such models. In particular, we discuss multi-population MFGs, graphon MFGs, major-minor MFGs, and Stackelberg MFGs, as well as variants involving cooperative players.