New Directions in Mean Field Games: MFG Subpopulation Behaviours and Graphon-MFG Systems

A new form of Mean Field Games (MFGs) called Graphon Mean Field Games (GMFGs) is developed and analysed in this program of research together with closely associated theories and their methodologies. GMFGs are non-cooperative games defined for large sub-populations of agents concentrated on the nodes of large, heterogeneous, networks and their asymptotic (graphon) limits. In this setting, game theoretic Nash equilibria are expressed in terms of the newly defined GMFG PDE equations, which are a sweeping generalization of the classical MFG PDEs. A key step in the project is the completion of the rigorous derivation of the central theoretical results of GMFG theory. In a closely related development, the theory of graphons is used as a basis for Graphon Linear Control (GLC) theory in which large populations of centrally controlled dynamical systems which interact over large, heterogeneous networks are analyzed and optimized in a tractable way. Specifically, the project includes the generation of a Graphon Linear Control (GLC) theory for the control of large populations of dynamical linear system agents with quadratic performance functions on large networks. In particular, an approximation theory of associated numerical methods is established based upon finite dimensional invariant subspaces of the system graphon operator. This program is currently being extended for GLC systems subject to network-wide Q-noise. Finally, a feature of the research program has been the development of the theory of GMFG systems embedded in Euclidean space; this underlies the variational analysis of agent performance in networks as a function of network position, where this theory includes sparse as well as dense networks.