Graphon Mean Field Games and the GMFG Equations

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.

37 pages; an abbreviated version has appeared in SIAM J. Control Optim

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• An Overview of Some Extensions of Mean Field Games beyond Perfect Homogeneity and Anonymity
• The Shortest-Path distance on graphons
• Capturing the critical coupling of large random Kuramoto networks with graphons
• Persistence of steady-states for dynamical systems on large networks