The lecture focused on a new theory, developed in collaboration with Mr. Jean-Michel Lasry, called the theory of " mean-field games ". The aim of this theory is to mathematically model (and analyze these models) situations involving a very large number of rational players (in the sense of Economics, i.e. optimizing their decisions and actions), each player interacting with the others " en moyenne " i.e. basing their choices or decisions on the average behavior of the other players. This type of situation is naturally encountered in Economics and Finance, where each player-agent optimizes his or her actions on the basis of global information, i.e. averaged over all players.
Other fields of application include transportation and traffic studies, or biology and ecology. More specifically, we consider Nash equilibria with N players, extend N to infinity to derive equilibrium models for continua of players, and analyze the resulting systems of nonlinear Partial Differential Equations (PDEs for short). The equations we introduce in this way are very general and contain, as special cases, many other classical equations such as the semi-linear elliptic equations, the Hartree-type equations of Quantum Mechanics, the Euler equations of Fluid Mechanics, kinetic models such as the Vlasov or Boltzmann equations, the equations of optimal mass transport (Monge-Kantorovich problem) or the Euler-Lagrange equations associated with optimal PDE control problems..
The terminology " mean field " comes from Physics and Mechanics, and it's not just a simple analogy, since our approach does indeed contain as a special case the classical mean field theories in Physics or Mechanics (see the examples cited above). Somewhat vaguely, a special case of our theory is where the players no longer have a choice and are then passively subjected to interactions with the rest of the players like physical " particles " or matter elements in Continuum Mechanics.
This means that the class of mathematical models obtained by this modeling approach is extremely vast, and raises a host of Mathematical Analysis questions, many of which have yet to be resolved.
Finally, let's point out an important extension of the theory as we have presented it so far. Implicitly, in all the foregoing, we are considering a homogeneous set made up of a very large number of identical players. It is often useful and realistic to introduce several categories of players (or agents, or living organisms...), each category being made up of a very large number of identical players
of identical players. What's more, the total number of players (population size) can vary over time (birth and death, attribute exchanges...).
The systems we introduce in these cases then contain, as special cases, models of population dynamics or chemical reactions..
This year's lecture was a kind of introduction to this new theory, focusing on the most elementary example of stationary problems for simple avoidance/grouping games.