Finite State Graphon Games with Applications to Epidemics

We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented with a graphon, which can be viewed as the limit of a dense random graph. A player's transition rates between the states depend on their control and the strength of interaction with the other players. We develop a rigorous mathematical framework for the game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing Nash equilibria. Moreover, we propose a numerical approach based on machine learning methods and we present experimental results on different applications to compartmental models in epidemiology.