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arXiv · 2411.07989

Convergence Analysis and Acceleration of Fictitious Play for General Mean-Field Games via the Best Response

Jiajia Yu·Xiuyuan Cheng·Jian-Guo Liu·Hongkai Zhao
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Abstract

A mean-field game (MFG) seeks the Nash Equilibrium of a game involving a continuum of players, where the Nash Equilibrium corresponds to a fixed point of the best-response mapping. However, simple fixed-point iterations do not always guarantee convergence. Fictitious play is an iterative algorithm that leverages a best-response mapping combined with a weighted average. Through a thorough study of the best-response mapping, this paper develops a simple and unified convergence analysis, providing the first explicit convergence rate for the fictitious play algorithm in MFGs of general types, especially non-potential MFGs. We demonstrate that the convergence and rate can be controlled through the weighting parameter in the algorithm, with linear convergence achievable under a general assumption. Building on this analysis, we propose two strategies to accelerate fictitious play. The first uses a backtracking line search to optimize the weighting parameter, while the second employs a hierarchical grid strategy to enhance stability and computational efficiency. We demonstrate the effectiveness of these acceleration techniques and validate our convergence rate analysis with various numerical examples.

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Jiajia Yu, Xiuyuan Cheng, Jian-Guo Liu, Hongkai Zhao. 2024-11-12. Convergence Analysis and Acceleration of Fictitious Play for General Mean-Field Games via the Best Response. https://arxiv.org/abs/2411.07989

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