Study of some models from game theory in the medium field.

Summary Mean-field game theory is a powerful formalism recently introduced to study stochastic optimization problems with a large number of agents. After recalling the basic principles of this theory and presenting some typical applications, we study in detail a stylized model of a seminar, of the mean field type. We derive an exact equation that allows us to predict the start time of the seminar and analyze different limit regimes, in which we arrive at approximate expressions of the solution. Thus we obtain a "phase diagram" of the problem. We then turn to a more complex population model with attractive group effects. Thanks to a formal analogy with the nonlinear Schrödinger equation, we show general evolution laws for the mean values of the problem, that the system verifies certain conservation laws and we develop approximations of variational type. This allows us to understand the qualitative behavior of the problem in the regime of strong interactions.