Mathematics

Mean Field Games (MFG) systems have been introduced simultaneously in 2006 by Lasry-Lions and Huang-Caines-Malhamé to describe Nash equilibria in differential games with infinitely many players. MFG models are the simplest "stochastic control"-type models taking into account interactions between rational agents. Their importance is also reflected by the fact that they are used to model systems that belong to very different areas such as economics, finance, social sciences and engineering.  After introducing the model we show how the existence and uniqueness of classical solutions can be proved using a fixed point theorem in some simple cases. However, to deal with first order and second order degenerate Mean Field Games variational techniques are necessary. The idea consists in characterizing weak solutions as minimizers of two optimal control problems in duality. This strategy was introduced for the Monge-Kantorovich mass transfer problem by Benamou and  Brenier, extended by Carlier, Cardaliaguet and Nazaret and applied to MFG by Cardaliaguet and collaborators. We discuss how these techniques can be applied to solve a game in which a central planner would like to steer a population to a predetermined final configuration, while still allowing individuals to choose their own strategies. This problem is known as the planning problem in MFG and can be seen as an optimal transport problem. The strong convexity present in the problem implies then Sobolev regularity estimates for the distributional solutions.