First-order mean-field games on networks and Wardrop equilibrium

TL;DR

This paper explores the relationship between Wardrop equilibrium models and stationary first-order mean-field games on networks, establishing a reformulation that allows solutions of one to inform the other and analyzing cost properties and calibration methods.

Contribution

It introduces a novel reformulation linking MFGs and Wardrop equilibria on networks, and provides methods for calibration and analysis of cost properties.

Findings

MFG solutions can be recovered from Wardrop problem solutions.

Reformulation establishes equivalence between MFG and Wardrop models.

Simple travel costs can lead to non-monotone MFGs.

Abstract

Here, we examine the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary mean-field games (MFG). In the first part of this paper, we present the Wardrop and the first-order MFG models on networks. Then, we show how to reformulate the MFG problem into a Wardrop problem and prove that the MFG solution is the Wardrop equilibrium for the corresponding Wardrop problem. Moreover, we prove that the solution of the MFG problem can be recovered using the solution to the associated Wardrop problem. Finally, we study the cost properties and the calibration of MFG with Wardrop travel cost problems. We describe a novel approach to the calibration of MFGs. Further, we show that even simple travel costs can give rise to non-monotone MFGs.