Speaker
Description
Mean Field Games (MFG) are models for Nash equilibria in games of optimal control involving a continuum of players. MFG are typically characterized by a system of PDEs which describes the nonlinear relationship between the optimal actions of individual players and the distribution of players in the game. The MFG system is given by a Hamilton—Jacobi—Bellman (HJB) equation, for the generic player’s control optimization, that is coupled nonlinearly with a Kolmogorov—Fokker—Planck (KFP) equation that determines the player density. To model MFG where players have access to possibly non-unique optimal controls and the Hamiltonian is nondifferentiable, we recently proposed that the MFG PDE system be relaxed to a Partial Differential Inclusion (PDI) system based on measurable selections of the partial subdifferential of the Hamiltonian.
In this talk, we study second-order MFG PDI with convex, Lipschitz continuous, but possibly nondifferentiable Hamiltonians, and their approximation by systems of classical MFG PDE with regularized Hamiltonians. Under broad conditions on the problem data, we prove that, up to subsequences, the solutions of the regularized problems converge to solutions of the MFG PDI. In particular, we show the convergence of the value functions in the H1-norm and of the densities in Lq-norms. Under stronger hypotheses on the problem data, we also establish rates of convergence between the solutions of the original and regularized problems, without requiring any higher regularity of the solutions. We give concrete examples that demonstrate the sharpness of several aspects of the analysis.