Abstract

I will give a brief introduction to the the emerging topic of Mean Field Games, introduced some years ago (by two groups at the same time: Huang, Malhamé and Caines, but also Lasry and Lions, who then devoted 6 subsequent editions of his course at Collège de France to its mathematical analysis) 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 cost. 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 Hamilton-Jacobi 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. These results are also needed to make rigorous the connection between optimality and equilibrium and allow for an interesting extension where instead of penalizing higher densities we insert a density constraint. Globally, the talk will be based on papers in collaborations with P. Cardaliaguet, A. Mészáros, J.-D. Benamou, G. Carlier, A. Prosinski and H. Lavenant.