Probabilistic Theory of Mean-Field Games

Lecture 1. The Two Pronged Probabilistic Approach to Mean Field Stochastic Differential Games
(Tuesday June 21, 14:30-16:30, LT 130, Huxley Building)

We review the paradigm of Mean Field Games introduced by Lasry and Lions, and independently by Caines Huang and Malhame under the name of Nash Certainty Equivalent. We introduce two variants of a probabilistic approach based on the solution of Backward Stochastic Differential Equations. For each of them, we give (without complete proofs) sufficient conditions for the existence of solutions, and we provide examples of applications. These applications are chosen from problems already discussed in the existing literature, but for which no mathematical solution was given. Among other examples, we shall discuss macro-economic growth models (Aiyagari & Krusell-Smith), flocking from generalized Cucker-Smale type models, and exit from a room in the presence of congestion.

Lecture 2. Calculus over Wasserstein Space and the Master Equation
(Wednesday June 22, 17:00-19:00, LT 340, Huxley Building)

We review some of the classical properties of spaces of probability measures at the core of the theory of optimal transportation. We concentrate on the notion of differentiability of functions of measures, and we recast the notion of differentiability introduced by P.L. Lions in this context. We use this notion of differentiability, which we call L-differentiability, to derive a couple of Ito formulae for functions of the form f(t;Xt; _t) where Xt is an Ito process in Rd and _t = L(_t) is the marginal distribution at time t of a possibly different Ito process. Using the chain rules provided by these It^o formulae, we show how one can derive the master equations touted in the theory of meanfield games and the optimal control of McKean-Vlasov stochastic differential equations.

Lecture 3. Applications to Games with Major and Minor Players, Finite State Space Games and Games of Timing
(Thursday June 23, 10:00-12:00, LT 144, Huxley Building)

Still in the framework of stochastic differential games, we introduce a formalism to capture the features of games with major and minor players. Without presenting full solutions, we highlight the McKean-Vlasov nature of some of the problems arising from the search of best response functions. Next, motivated by applications to health sciences, social sciences, cyber security, … , we consider Mean Field Games with finite state spaces and recover most of the results of the classical theory of stochastic differential mean field games, including the master equation and the analysis of games with major and minor players. Finally, after the presentation of a simple diffusion model for the banking system and bank runs, we introduce a formalism to account for games of timing and we propose several solutions depending upon the nature of the assumptions which can be made on the coefficients.