Graphon Mean Field Games and the GMFG Equations

TL;DR

This paper rigorously establishes the existence, uniqueness, and epsilon-Nash equilibrium properties of Graphon Mean Field Game (GMFG) equations, advancing the theoretical understanding of decentralized control on large networks.

Contribution

It provides rigorous proofs for the existence, uniqueness, and epsilon-Nash equilibria of GMFG equations, extending prior formulations to a solid theoretical foundation.

Findings

Proved existence of solutions to GMFG equations

Established uniqueness of GMFG solutions

Demonstrated epsilon-Nash equilibrium properties

Abstract

The emergence of the graphon theory of large networks and their infinite limits has enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks (Gao and Caines, IEEE CDC 2017, 2018). Furthermore, the study of the decentralized control of such systems was initiated in (Caines and Huang, IEEE CDC 2018, 2019), where Graphon Mean Field Games (GMFG) and the GMFG equations were formulated for the analysis of non-cooperative dynamic games on unbounded networks. In that work, existence and uniqueness results were introduced for the GMFG equations, together with an epsilon-Nash theory for GMFG systems which relates infinite population equilibria on infinite networks to finite population equilibria on finite networks. Those results are rigorously established in this paper.