Finite State Graphon Games with Applications to Epidemics

Abstract

We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented by a graphon, which can be viewed as the limit of a dense random graph. The player's transition rates between the states depend on their own control and the interaction strengths with the other players. We develop a rigorous mathematical framework for this 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 equilibria. Moreover, we propose a numerical approach based on machine learning tools and show experimental results on different applications to compartmental models in epidemiology.

Learning Graphon Mean Field Games and Approximate Nash Equilibria

“…There have since been efforts to control cooperative graphon mean field systems with diffusive linear dynamics using spectral methods (Gao & Caines, 2019a;b). On the other hand, Bayraktar et al (2020); Bet et al (2020) consider large non-clustered systems in a continuous-time diffusion-type setting without control, while Aurell et al (2021b) and Aurell et al (2021a) consider continuous-time linear-quadratic systems and continuous-time jump processes respectively. To the best of our knowledge, only Vasal et al (2021) have considered solving and formulating a graphon mean field game in discrete time, though requiring analytic computation of an infinite-dimensional value function defined over all mean fields and thus being inapplicable to arbitrary problems in a black-box, learning manner.…”