Variational mean field games

Abstract

This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e., with diffusion) is also presented in both the stationary and dynamic cases. The variational problems relevant to MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on augmented Lagrangian algorithms, and some new simulations are shown at the end of the paper.