Contributions to medium field game theory.

Summary This thesis deals with the study of new medium field game models. We first study optimal stopping and impulse control models in the absence of common noise. We construct for these models a notion of adapted solution for which we prove existence and uniqueness results under natural assumptions. Then, we focus on several properties of mean-field games. We study the limit of these models to pure evolution models when the players' anticipation tends to 0. We show the uniqueness of equilibria for strongly coupled systems (coupled by strategies) under certain assumptions. We then prove some regularity results on a master equation that models a mean field game with common noise in a discrete state space. We then present a generalization of the standard Uzawa algorithm and apply it to the numerical solution of some mean-field game models, in particular optimal stopping or impulse control. Finally, we present a concrete case of mean-field game that comes from problems involving a large number of connected devices in telecommunications.