Selected topics in transport phenomena: deterministic and probabilistic aspects

Mean Field Games: theory and applications

René Carmona

Princeton University

We review the Mean Field Game paradigm introduced independently by Caines-Huang-Malhame and Lasry-Lyons ten years ago, and we illustrate their relevance to applications with a few practical of examples (bird flocking, room exit, systemic risk, cyber-security, .... ). We then review the probabilistic approach based on Forward-Backward Stochastic Differential Equations, and we derive the Master Equation from a version of the chain rule (Ito's formula) for functions over flows of probability measures. Finally, we present a (possibly) new formulation of the mean field game problem in the presence of major and minor players, and give new existence results for linear quadratic models and models with finite state spaces. We shall also provide numerical results illustrating the theory and raising new challenging open problems.