Mean Field Games under displacement monotonicity: well-posedness and a bit of numerics
Mean Field Games (MFGs) is a mathematical framework to describe strategic decision making in large populations of interacting agents. This theory was initiated about two decades ago by Lasry—Lions on the one hand and by Huang—Malhamé—Caines on the other hand. Such problems can be described by systems of coupled nonlinear PDEs: a Hamilton—Jacobi—Bellman equation describing the value function of a typical agent and a Kolmogorov—Fokker—Planck equation describing the evolution of the agents’ density. In this talk we will present some recent developments on such PDE system when the data satisfy a suitable notion of monotonicity condition. In contrast with the widely used notion of so-called Lasry—Lions monotonicity, we consider the so-called displacement monotonicity regime, which stems from the theory of optimal mass transport. We will show how such conditions give new tools to tackle problems with degenerate noise involving a large class of Hamiltonians. In the second half of the talk we will demonstrate how such monotonicity conditions could be used, beyond the well-posedness theory, to build efficient and stable numerical schemes for first order models, where we cannot rely on the regularisation effect of non-degenerate noise.