Selected topics in mean field games

Abstract

Mean field game theory was initiated a little more than 15 years ago with the aim of simplifying the search for Nash equilibria in games with a large number of weakly interacting players. Since then, a lot has been done. Numerous equilibrium existence results have been obtained, using different characterizations and in various contexts. The analysis of the master equation, which describes the evolution of the value of the game, has also seen significant progress, which has, for example, allowed establishing in certain cases the convergence of games with a finite number of players. However, mean field games remain of a complex nature. For instance, the typical lack of uniqueness of solutions raises selection issues that are still poorly understood. The objective of the note is to present some of the latest advances, as well as some avenues to address further challenging questions.