Carla Crucianelli (Princeton): Non-linear graphon games

When studying a non-cooperative stochastic differential game, it is classical to use the mean-field game framework. However, this approach has its limitations when the particles interact through a random graph. This is because we expect the heterogeneity of the system to be retained in the large population limit. In this talk, we consider a non-cooperative stochastic differential game with player interactions modeled through a sequence of graphs converging to a graphon. We show that under standard regularity conditions and dissipativity of the drift of the state process, when the number of players goes to infinity, the game converges to a general graphon game. The limit is quantitative and based on a new form of propagation of chaos for interacting FBSDEs on graphs. Further, when the cost function satisfies the Lasry-Lions monotonicity condition, we propose an existence and uniqueness result for graphon games. Our proof relies on investigating a system of uncountably many McKean-Vlasov type BSDEs driven by a continuum of Brownian motions. We study this BSDE in a specific probability framework called a Fubini extension and discuss its well-posedness, characterize the graphon mean-field equilibrium and show its existence and uniqueness. The existence proof is based on a new application of the continuation method. Based on joint works with Dylan Possamaï and Ludovic Tangpi.