An Overview of Some Extensions of Mean Field Games beyond Perfect Homogeneity and Anonymity

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.