Mean-Field Games Through Monotone Methods
In this talk, we address the study of mean-field games (MFGs) using monotonicity techniques. MFGs model the limit of differential games with a large population as the number of agents tends to infinity. In these models, each agent is rational and optimizes a cost functional, and often comprise a system of two equations, a Hamilton–Jacobi equation and a Fokker–Planck equation, which can be associated with a monotone operator. This structure is key to establish the uniqueness of solutions, as used by Lasry and Lions. A main difficulty in the MFG theory is to establish the existence of solutions. While Hamilton–Jacobi and Fokker–Planck equations are extremely well understood, the coupling between these two equations presents substantial difficulties. Our main purpose is to discuss how monotone operators ideas enable a unified approach to address existence of weak solutions to a wide class of MFGs, including stationary problems with periodic or Dirichlet boundary conditions, time-dependent problems, and planning problems with congestion.
