Heterogeneous Mean Field Games and Local Well-posedness
Bixing Qiao
TL;DR
The paper develops Heterogeneous Mean Field Games (HMFG) to model asymmetric populations interacting through a density ensemble, unifying Graphon MFGs within an infinite-dimensional FBSDE framework. It constructs a rigorous Fubini-extension and a complete metric space of measure-valued population laws, derives an Itô formula on measure flows, and formulates HMFG as a fixed-point problem over population laws. Local well-posedness of the infinite-dimensional FBSDE yields a unique HMFG equilibrium, and a propagation-of-chaos result shows this equilibrium serves as a good approximation for large $N$-player games. The master equation is obtained as the decoupling field of the FBSDE, establishing a pathway to dynamic value characterizations, with global well-posedness left for future work.
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 Itô 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.