Infinite horizon LQG Graphon Mean Field Games: Explicit Nash values and local minima

In this study, we generalize the analysis of infinite horizon linear quadratic Gaussian (LQG) Mean Field Games within the framework of Graphon Mean Field Games (GMFG) introduced in Caines and Huang (2018) over finite horizons. Graphon Mean Field Games (GMFGs) are non-uniform generalizations of Mean Field Games where the non-uniformity of agents is characterized by the nodes on which they are located in a network. Under mild assumptions on the structure of the network and parameters of the game, we obtain for almost every node, an explicit analytical expression for the Nash values (i.e. the cost at equilibrium). With additional assumptions, we provide sufficient conditions for nodes to have locally minimal Nash values. We illustrate the results for the uniform attachment network.