Graphon Mean Field Games and the GMFG Equations

The existence of Nash equilibria in the Mean Field Game (MFG) theory of large non-cooperative populations of stochastic dynamical agents is established by passing to the infinite population limit. Individual agent feedback strategies are obtained via the MFG equations consisting of (i) a McKean-Vlasov-Hamilton-Jacobi-Bellman equation generating the Nash values and the best response control actions, and (ii) a McKean-Vlasov-Fokker-Planck-Kolmogorov equation for the probability distribution of the state of a generic agent in the population, otherwise known as the mean field. The applications of MFG theory now extend from economics and finance to epidemiology and physics.

In current work, MFG and MF Control theory is extended to Graphon Mean Field Game (GMFG) and Graphon Mean Field Control (GMFC) theory. Very large scale networks linking dynamical agents are now ubiquitous, with examples being given by electrical power grids, the internet, financial networks and epidemiological and social networks. In this setting, the emergence of the graphon theory of infinite networks has enabled the formulation of the GMFG equations for which we have established the existence and uniqueness of solutions. Applications of GMFG and GMFC theory to systems on particular networks of interest are being investigated and computational methods developed. As in the case of MFG theory, it is the simplicity of the infinite population GMFG and GMFC strategies which, in principle, permits their application to otherwise intractable problems involving large populations on complex networks. Work with Minyi Huang