Finite State Graphon Mean Field Games

Abstract:

In this work we will talk about stochastic games on large graphs, where the players no longer interact with each other symmetrically. To encode this information, the concept of graphon is employed. Graphons are the natural continuum limits for dense interaction matrices.
Mathematically, a graphon is a symmetric measurable function $W : [0, 1]^2 \to [0,1]$, with $W(u, v)$ representing the interaction between vertices players $u$ and $v$.
We consider continuous-time controlled dynamics on finite states: we write the dynamics of the $N$-player game as a system of stochastic differential equations driven by independent stationary Poisson random measures. Under a set of fairly general assumptions, we derive the existence of both relaxed and nonrelaxed controls, while uniqueness is proven under the Lasry-Lions monotonicity conditions.
This talk is based on joint research with Luca Di Persio (University of Verona, Italy) and Francesco Giuseppe Cordoni (University of Trento, Italy).