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

TL;DR

This paper addresses ambient potential identification in mean-field games by recovering an environment-dependent potential $q$ from measurements of the Nash equilibrium via the initial value function $\phi_0$. It introduces Equilibrium Correction Iteration (ECI), with acceleration variants Best Response Iteration (BRI) and Hierarchical ECI (HECI), leveraging the forward MFG structure rather than generic optimization. The authors establish a fixed-point-like update inspired by the HJB/FP coupling and demonstrate through extensive numerics that $\phi_0$ encodes sufficient information to reconstruct $q$ accurately, even under large viscosity $\nu$, with convergence evidence and practical speedups over optimization-based approaches. They also connect the ambient potential ID problem to inverse linear parabolic equations via Hopf-Cole transformation and show that discretization, grid hierarchy, and monotonicity of costs critically affect well-posedness and convergence. The results suggest robust, scalable methods for inverse MFG problems and point to future theoretical analysis of convergence and noise-robustness, with potential for broader applicability to related inverse PDE problems.

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.

Figures (10)

• Figure 1: ECI efficacy with 1D example (§\ref{['subsec: num effective']}). The measurement-induced term $M(\phi_0):=\frac{1}{T}\int_{\mathbb{T}^d}\phi_0(x)\mathrm{d} x-\nu\Delta\phi_0+H(\nabla\phi_0)$ retains substantial information about the true potential $q$. The ECI algorithm utilizing $M(\phi_0)$ rapidly recovers the main structure of the true potential.
• Figure 2: ECI efficacy with 2D example (§ \ref{['subsec: num effective']}). The measurement-induced term $M(\phi_0):=\frac{1}{T}\int_{\mathbb{T}^d}\phi_0(x)\mathrm{d} x-\nu\Delta\phi_0+H(\nabla\phi_0)$ retains substantial information about the true potential $q$. The ECI algorithm utilizing $M(\phi_0)$ rapidly recovers the main structure of the true potential.
• Figure 3: Scalability with respect to grid (§ \ref{['subsec: num comp ECI policy']}). The computational time is approximately linear in $n_x$, and the relative error improves as $n_x$ increases.
• Figure 4: Comparison of ECI and BRI (§ \ref{['subsec: num comp']}). The plots show the errors of recovered potential $q^{(K)} - q$ for BRI1(0.2) (top left), BRI1(0.5) (top right), BRI5(0.5) (bottom left), and ECI (bottom right). Starting from the same initial guess $q^{(0)} = 0$, all methods recover the true potential pointwise with a relative error less than $10^{-6}$.
• Figure 5: Comparison of ECI and BRI (§ \ref{['subsec: num comp']}). The top row shows the relative error of the potential $q^{(k)}$ (left) and of the measurement $\phi_0^{(k)}$ (center) and the residue of the forward solver (right) versus the number of outer loops. The bottom row incorporates the cost of the inner loop iterations and plots the relative error of the potential $q^{(k)}$ (left) and the measurement $\phi_0^{(k)}$ (center) versus the number of HJB/FP solves. Each marker of ECI corresponds to an outer loop iteration. BRI1 and BRI5 achieve a similar relative error as ECI in a comparable number of outer loops but with fewer calls to the HJB and FP solvers.

Theorems & Definitions (6)

• Proposition 2.2
• Remark 2.4
• Proposition 2.5
• Proposition 3.1
• Proposition 4.1: Hopf-Cole Transformation for MFGs gueant2011meangueant2012hopfcole
• proof