This 2-day short course is designed to introduce attendees to the rapidly developing and important area of Mean Field Games. Much of the research activities in Mean Field Games is taking place in Europe. The IMA is pleased to offer an intense program to introduce this topic in the US.
Mean Field Games, which models the the dynamics of large number of agents, has applications in many areas such as economics, finance, dynamics of crowds as well as in biology and and social sciences. The starting point is the analysis of N-player differential games when N tends to infinity. Usually, differential games with N-players can be summed up by a Hamilton-Jacobi system of PDEs. Unfortunately, for large values of N, this system turns out to be untractable. The key point of Mean Field Game theory is that things do simplify, at least for a wide range of games that are symmetrical with respect to the players, as the number of players increases.
The theory of continua of players is not new in the literature and has been widely used since Robert Auman's seminal paper on general equilibrium with infinitely many players. However, the Mean Field Game approach is different in many ways from what has been studied. The most striking particularity is the kind of equations describing the limit system in which each player has become infinitesimal amidst the mass of other players.
There are two ways to formulate this system. The most general one involves a nonlinear second order partial differential equation in the space of probability measures. The analysis of this equation is mathematical challenging and the source of a wide number of questions. For instance a finite dimensional analogue of this equation is a new nonlinear system of transport equations, which enjoys astonishing properties: it is monotonicity preserving; its solutions can (in some particular cases) be obtained as the gradient of solutions of Hamilton-Jacobi equations, etc. More surprisingly, the analysis of this system sheds new lights on more standard Hamilton-Jacobi equations.
As for the second approach, one way to derive it is by analyzing the characteristics of the general equation (which can be defined for a large class - but not all - games). This gives rise to a (new) finite dimensional system of PDEs, coupling a Hamilton-Jacobi equation (describing the behavior of a typical agent) with a Kolmogorov equation (formalizing the evolution of the agents' density). Here again the mathematical structure is very rich: in particular this is a Hamiltonian system, which can be interpreted in terms of optimal control of Hamilton-Jacobi or of Kolmogorov equations.
The goal of the short course is to give the participants a basic introduction to the subject of Mean Field Games and to present to them the most recent developments in the theory, computations, and applications. Mean Field Games is an important research area that deserves the attention of the broader mathematical community.
Yves Achdou, Université de Paris VII (Denis Diderot)
Pierre Cardaliaguet, Université de Paris-Dauphine
Pierre-Louis Lions, College de France