A Policy Iteration Method for Inverse Mean Field Games

TL;DR

This paper introduces a policy iteration approach for efficiently solving inverse mean-field game problems by reconstructing obstacle functions from partial data, with proven convergence and demonstrated superior performance.

Contribution

A novel policy iteration method for inverse MFGs that decouples the complex inverse problem into linear PDE solutions, improving efficiency and accuracy.

Findings

Proven linear convergence rate of the method.

Numerical examples show superior accuracy over direct least-squares methods.

Method effectively reconstructs obstacle functions from partial observations.

Abstract

We propose a policy iteration method to solve an inverse problem for a mean-field game (MFG) model, specifically to reconstruct the obstacle function in the game from the partial observation data of value functions, which represent the optimal costs for agents. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. This method can also be viewed as a fixed-point iteration that simultaneously solves the MFG system and inversion. We prove its linear rate of convergence. In addition, numerical examples in 1D and 2D, along with performance comparisons to a direct least-squares method, demonstrate the superior efficiency and accuracy of the proposed method for solving inverse MFGs.