Examples of uniqueness restoration and equilibrium selection in medium field games.

Summary The objective of this manuscript is to present several results on uniqueness restoration and equilibrium selection in mean field games. The theory of mean-field games was initiated in the 2000s by two groups of researchers, Lasry and Lions in France, and Huang, Caines and Malhamé in Canada. The objective of this theory is to describe Nash equilibria in stochastic differential games including a large number of players interacting with each other through their common empirical measure and presenting sufficient symmetry. While the existence of equilibria in mean-field games is now well understood, the uniqueness remains known in a very limited number of cases. In this respect, the best known condition is the so-called monotonicity condition, due to Lasry and Lions. In this thesis, we show that, for a certain class of mean-field games, uniqueness can be restored using a random forcing of the dynamics, common to all players. Such a forcing is called "common noise". We also show that, in some cases, it is possible to select equilibria in the absence of common noise by making the common noise tend to zero. Finally, we show how these results apply to principal-agent problems, with a large number of interacting agents.