A Variational Approach For Price Formation Models In One Dimension

TL;DR

This paper introduces a variational method for solving one-dimensional price formation models modeled by mean-field games, simplifying the system and providing a new numerical approach with proven existence of solutions.

Contribution

It develops a variational formulation for first-order mean-field game models of price formation, enabling more efficient numerical solutions and establishing solution existence.

Findings

Variational problem simplifies the original MFG system.

Numerical results demonstrate the method's effectiveness.

Existence of solutions is proven using calculus of variations.

Abstract

In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincar{\'e} Lemma, we eliminate one of the equations and obtain a variational problem for a single function. This variational problem offers an alternative approach for the numerical solution of the original MFGs system. We show a correspondence between solutions of the MFGs system and the variational problem. Moreover, we address the existence of solutions for the variational problem using the direct method in the calculus of variations. We end the paper with numerical results for a linear-quadratic model.