Equilibrium Correction Iteration for A Class of Mean-Field Game Inverse Problem

TL;DR

This paper introduces the Equilibrium Correction Iteration (ECI), an iterative method for recovering unknown potentials in inverse Mean-Field Games, with acceleration variants and insights into the problem's theoretical connections and numerical behavior.

Contribution

The paper proposes ECI, a structure-exploiting iterative approach for inverse MFGs, along with acceleration techniques and theoretical insights linking to inverse linear parabolic equations.

Findings

ECI effectively uncovers hidden potential information from equilibrium data.

Acceleration variants BRI and HECI improve computational efficiency and accuracy.

Numerical experiments reveal factors affecting the inverse problem's well-posedness.

Abstract

This work investigates the ambient potential identification problem in inverse Mean-Field Games (MFGs), where the goal is to recover the unknown potential from the value function at equilibrium. We propose a simple yet effective iterative strategy, Equilibrium Correction Iteration (ECI), that leverages the structure of MFGs rather than relying on generic optimization formulations. ECI uncovers hidden information from equilibrium measurements, offering a new perspective on inverse MFGs. To improve computational efficiency, two acceleration variants are introduced: Best Response Iteration (BRI), which uses inexact forward solvers, and Hierarchical ECI (HECI), which incorporates multilevel grids. While BRI performs efficiently in general settings, HECI proves particularly effective in recovering low-frequency potentials. We also highlight a connection between the potential identification problem in inverse MFGs and inverse linear parabolic equations, suggesting promising directions for future theoretical analysis. Finally, comprehensive numerical experiments demonstrate how viscosity, terminal time, and interaction costs can influence the well-posedness of the inverse problem.