Heterogeneous Mean Field Games and Local Well-posedness

TL;DR

This paper introduces a comprehensive framework for Heterogeneous Mean Field Games (HMFG), demonstrating local well-posedness, uniqueness of equilibrium, and its approximation properties for N-Player Games, along with deriving the master equation.

Contribution

It develops a general HMFG framework that includes graphon mean field games, proves local well-posedness, and derives the master equation for the infinite-dimensional system.

Findings

HMFG system is an infinite-dimensional FBSDE.

Unique equilibrium exists for HMFG.

Equilibrium approximates N-Player Game outcomes.

Abstract

Motivated by the recent interests in asymmetric mean field games, this paper provides a general framework of Heterogeneous Mean Field Game (HMFG) that subsumes different formulations of graphon mean field games. The key feature of the HMFG is that the players interact with the population through the density ensemble. In this case, the HMFG system becomes an infinite-dimensional Forward-Backward SDE (FBSDE) system. We show that the FBSDE is locally well-posed, thus the HMFG has a unique equilibrium. In addition, we show that the equilibrium of HMFG is a good approximate equilibrium of the corresponding N-Player Game. Lastly, we derive the It\^{o} formula of infinite-dimensional measure flow and use it to obtain the master equation for HMFG as a decoupling field of the infinite-dimensional FBSDE system.