PDE seminar, January 26th 2016
Tuesday 26th EHESS, 190198 Avenue de France, room 466, 4th floor  75013 Paris
Poster here

Variational Mean Field Games
I will give a brief introduction to the the emerging topic of Mean Field Games, introduced by JM Lasry and PL Lions some years ago as a model for the equilibrium of a population of agents each selecting his own optimal paths, according to a criterion which involves the density of the other agents, in the form of a congestion charge. This gives rise to a coupled system of PDEs, a continuity equation where the density moves according to the gradient of a value function, and a HamiltonJacobi equation solved by the value function, where the density also appears. I will mainly deal with the case where this equilibrium problem may be seen as optimality conditions of a convex variational problem, and give the main results in this framework. In particular, I will present some easy but recent regularity results, as well as the connection with optimal transport theory (in particular the dynamical formulation given by JD Benamou and Y Brenier for numerical purposes). At the end of the talk I will present an interesting variant, where the congestion cost is replaced by a capacity constraint. 