Monotonicity methods for Mean-Field Games

Student thesis: Doctoral Thesis

Abstract

Mean-field games (MFGs) model the behavior of large populations of rational agents. Each agent seeks to minimize an individual cost that depends on the statistical distribution of the population. Roughly speaking, MFGs are given by the limit of differential games with N agents as N goes to infinity. This limit describes an average effect of the population’s behavior. Instead of modeling large systems for all agents, we consider two coupled equations: the Hamilton–Jacobi equation and the Fokker–Planck equation. A solution to MFGs is given by two functions: a value function and a population density. From the point of view of mathematics, monotonicity conditions for MFGs are a natural way to obtain the uniqueness of solutions and the stability of systems. In this thesis, we develop a new framework to establish the existence of solutions to MFGs through monotonicity. First, we study first-order stationary monotone MFGs with Dirichlet boundary conditions. In MFGs, boundary conditions arise when agents can leave the domain. There are exit costs for agents given by Dirichlet boundary conditions. Here, we establish the existence of solutions to MFGs that fulfill those boundary conditions in the trace sense. In particular, our solution is continuous up to the boundary in the one-dimensional case. Second, we consider time-dependent monotone MFGs with space-periodic boundary conditions. To solve the time-dependent monotone MFG, we introduce a mono- tone high-order regularized elliptic problem in Rn+1, although the original MFG is a parabolic type. To preserve monotonicity, we need to determine the specific boundary conditions for the time variable. Then, we can apply our method of stationary MFGs to this regularization. In particular, we prove that a solution to the problem exists for any terminal time. Third, we investigate stationary MFGs with hypoelliptic operators that are degenerate differential operators. Those models arise from stochastic control problems with the Stratonovich integration. We study a hypoelliptic MFG with the standard quadratic Hamiltonian. Under standard assumptions, although there is no uniform elliptic condition in hypoelliptic operators, we verify that there is a unique solution to our hypoelliptic MFG.
Date of Award Nov 22 2021 English (US) Computer, Electrical and Mathematical Sciences and Engineering Diogo Gomes (Supervisor)

Keywords

• mean-fields games
• monotone operators
• hypoelliptic operators

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