Middle-field games and variational inequalities

The theory of middle field games was proposed in 2006 in the works of J.M. Larry, P.-L. Lions and (independently) M. Huang, R. P. Malhame, P. Caines. The purpose of midfield games is to describe the collective behavior of a large group of similar agents (players) pursuing individual goals. Traditionally, the game of the middle field is reduced to a system of partial differential equations consisting of the Bellman equation and the Kolmogorov-Fokker-Planck equation. The Bellman equation defines the price function of each player, while the Kolmogorov-Fokker-Planck equation defines the dynamics of the distribution of players. In the talk, I will tell how the system of equations of the middle field game can be reduced to a variational inequality for measures over the Cartesian product of a time interval and a tangent bundle of phase space for one player. Using this approach, it is also possible to formulate the principle of dynamic programming for medium-field games.