Caltech/UCLA/USC Joint Analysis Seminar

UCLA, MS 6221

A Mean Field Game is a differential game (in the sense of game theory) where instead of a finite number of players we have a continuous distribution of (infinitely) many players, however we make the simplifying assumption that all players are identical. In this talk we consider the existence and uniqueness of Nash Equilibrium in Mean Field Games. We show why the study of Nash Equilibrium naturally leads to the study of a Hamilton-Jacobi equation over the space of measures called the master equation, whose solutions give rise to Nash Equilibrium for our game. For mean field games there isn't a general theory of viscosity solutions analogous to Hamilton-Jacobi equations in finite dimensions. Motivated by this we revisit the classical solution theory (as opposed to viscosity solutions) of Hamilton Jacobi equations and identify a symmetry that extends the well-posedness theory into new regimes. This symmetry also yields results for the master equation in mean field games. We will see that there are two natural types of noise that one can impose in a Mean Field Game, individual noise and common noise, which correspond to cases where the noise of each player is independent and identical respectively. Individual noise has a regularizing effect that is utilized in most well-posedness results for the master equation. We explore well-posedness for the master equation in the case without individual noise, under a monotonicity condition.