Existence and uniqueness result for mean field games with congestion effect on graphs

TL;DR

This paper establishes general conditions for the existence and uniqueness of solutions to mean field games on graphs, incorporating complex congestion effects that depend on player distribution across nodes.

Contribution

It introduces a broad existence and uniqueness framework for graph-based mean field games with complex congestion effects, extending previous discrete models.

Findings

Proved existence using a priori estimates and Schauder fixed point theorem.

Developed a new criterion for uniqueness with non-local Hamiltonian functions.

Generalized discrete mean field game results to more complex congestion scenarios.

Abstract

This paper presents a general existence and uniqueness result for mean field games equations on graphs ($\mathcal{G}$-MFG). In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument \`a la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of existing uniqueness results.