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We consider a game for a continuum of non-identical players evolving on afinite state space. Their heterogeneous interactions are represented by agraphon, which can be viewed as the limit of a dense random graph. The player'stransition rates between the states depend on their own control and theinteraction strengths with the other players. We develop a rigorousmathematical framework for this game and analyze Nash equilibria. We provide asufficient condition for a Nash equilibrium and prove existence of solutions toa continuum of fully coupled forward-backward ordinary differential equationscharacterizing equilibria. Moreover, we propose a numerical approach based onmachine learning tools and show experimental results on different applicationsto compartmental models in epidemiology.

Finite State Graphon Games with Applications to Epidemics