Mean Field Games: From Foundations to Singular Controls and Optimal Stopping

This short course provides an introduction to mean field games, with particular emphasis on their probabilistic formulation and on models involving singular control and optimal stopping. We will start from the basic idea of mean field interaction: a representative agent optimizes against the distribution of a large population, while equilibrium requires consistency between the agent’s optimal behavior and the induced population flow. After discussing this fixed point structure and some analytical tools, we will focus on classes of models motivated by economics and finance. Special attention will be devoted to mean field games in which agents make irreversible, monotone, or stopping decisions, both in stationary and non-stationary settings. In these models, a central theme is the interplay between free-boundary methods, reflected or absorbed diffusions, and fixed point arguments in the characterization of equilibria. We will also discuss how structural properties of the interaction, such as monotonicity or submodularity, can be exploited in suitable models to obtain comparison results and to gain insights into uniqueness or multiplicity of equilibria. The course will conclude by highlighting connections between mean field games and mean field control problems, as well as recent perspectives related to learning-based approaches.