Master Equation for Cournot Mean Field Games of Control with Absorption

TL;DR

This paper derives a unique solution to the master equation in a mean field game modeling Cournot competition with exhaustible resources, using advanced PDE analysis to handle absorption effects.

Contribution

It introduces a novel approach to prove existence and uniqueness of solutions for the master equation in control-based mean field games with absorption.

Findings

Established existence and uniqueness of the master equation solution.

Developed new a priori estimates for differentiability with respect to initial measures.

Analyzed a nonlocal Hamilton-Jacobi/Fokker-Planck system with boundary conditions.

Abstract

We establish the existence and uniqueness of a solution to the master equation for a mean field game of controls with absorption. The mean field game arises as a continuum limit of a dynamic game of exhaustible resources modeling Cournot competition between producers. The proof relies on an analysis of a forward-backward system of nonlocal Hamilton-Jacobi/Fokker-Planck equations with Dirichlet boundary conditions. In particular, we establish new a priori estimates to prove that solutions are differentiable with respect to the initial measure.