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.

翻译：本文提出一种策略迭代方法，用于求解平均场博弈（MFG）模型的逆问题，具体而言，即根据价值函数（代表智能体的最优成本）的部分观测数据重构博弈中的障碍函数。该方法将这一复杂的逆问题——一个受MFG中耦合的非线性前向-后向偏微分方程系统约束的优化问题——解耦为若干次线性偏微分方程求解与线性逆问题求解的迭代过程。此方法亦可视为同时求解MFG系统与逆问题的定点迭代。我们证明了该方法的线性收敛速率。此外，通过一维与二维数值算例，以及与直接最小二乘法的性能对比，验证了所提方法在求解平均场博弈逆问题上具有更优的效率和精度。