Existence and uniqueness of solutions for Bertrand and Cournot mean field games

TL;DR

This paper establishes the existence and uniqueness of classical solutions for a system of PDEs modeling Bertrand and Cournot competition among a continuum of resource producers, using new a priori estimates.

Contribution

It provides the first rigorous proof of existence and uniqueness for these mean field game models under broad conditions.

Findings

Proved existence of classical solutions under general assumptions

Established uniqueness under additional hypotheses

Developed new a priori estimates for the PDE system

Abstract

We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness. Keywords: mean field games, Hamilton-Jacobi, Fokker-Planck, coupled systems, optimal control, nonlinear partial differential equations