Discrete potential mean field games: duality and numerical resolution

TL;DR

This paper introduces a class of discrete mean field games with potential structure, analyzing their duality and proposing numerical methods for solving problems with congestion, price interactions, and hard constraints.

Contribution

It develops a duality framework for potential mean field games and introduces two novel numerical algorithms with convergence analysis.

Findings

Numerical methods successfully solve MFGs with hard constraints.

Duality approach links MFGs to optimal control problems.

Algorithms demonstrate convergence in example scenarios.

Abstract

We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows hard constraints on the distribution of the agents. We analyze the connection between the MFG problem and two optimal control problems in duality. We present two families of numerical methods and detail their implementation: (i) primal-dual proximal methods (and their extension with nonlinear proximity operators), (ii) the alternating direction method of multipliers (ADMM) and a variant called ADM-G. We give some convergence results. Numerical results are provided for two examples with hard constraints.