Graphon Mean Field Games and the GMFG Equations: $epsilon$-Nash Equilibria

Abstract

Very large networks linking dynamical agents are now ubiquitous and the need to analyse, design and control them is evident. 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, CDC 2017, 2018]. Moreover, the study of the decentralized control of such systems was initiated in [Caines and Huang, CDC 2018] 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 established for the GMFG equations, while the current work continues that analysis by developing an $epsilon$-Nash theory for GMFG systems by relating the infinite population equilibria on infinite networks to finite population equilibria on finite networks.