Non--exchangeable mean field games with moderate interactions and common noise

Abstract: We study mean field games for large non--exchangeable populations with moderate local interactions and common noise. The finite--player system is driven by two complementary interaction mechanisms : a graphon--type structure, which encodes heterogeneous large--scale interactions between agents, and a rescaled local kernel, which produces a density-dependent interaction term in the limit. The limiting model is a non--exchangeable mean field game in which the representative player is indexed by a label (u\in[0,1]), interacts through a graphon--weighted local density, and is affected by a graphon--induced environment law. We introduce a relaxed formulation of the limiting mean field game, adapted to the presence of common noise, and prove existence under general continuity and non--degeneracy assumptions. Under additional convexity assumptions, relaxed equilibria can be realized in strict form. In the deterministic case without common noise, we obtain deterministic equilibria and provide a probabilistic characterization of strict equilibria through a nonlinear Feynman--Kac representation. We then establish the asymptotic connection with the finite--player game. We prove that every limit point of approximate closed--loop Nash equilibria is a relaxed solution of the limiting mean field game, and that the corresponding averaged equilibrium payoffs converge. Conversely, every relaxed mean field game equilibrium can be approximated by Markovian approximate Nash equilibria of the finite--player systems. These results give a complete asymptotic characterization of equilibrium behavior for non--exchangeable games with moderate interactions and common noise.