Variational mean field games for market competition

TL;DR

This paper develops a mathematical framework for analyzing market competition models using mean field games, establishing existence, uniqueness, and regularity of solutions, and connecting these to convex optimization problems.

Contribution

It introduces a comprehensive analysis of Bertrand and Cournot mean field games with reflection boundaries, including new proofs of solution uniqueness and regularity.

Findings

Proved existence and uniqueness of solutions for the models.

Established the connection to convex minimization problems.

Extended results to the deterministic limit case.

Abstract

In this paper, we explore Bertrand and Cournot Mean Field Games models for market competition with reflection boundary conditions. We prove existence, uniqueness and regularity of solutions to the system of equations, and show that this system can be written as an optimality condition of a convex minimization problem. We also provide a short proof of uniqueness to the system addressed in [Graber, P. and Ben- soussan, A., Existence and uniqueness of solutions for Bertrand and Cournot mean field games, Applied Mathematics & Optimization (2016)], where uniqueness was only proved for small parameters $\epsilon$. Finally, we prove existence and uniqueness of weak solutions to the corresponding first order system at the deterministic limit.