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

TL;DR

This paper provides a comprehensive convergence analysis of fictitious play in general mean-field games, introduces acceleration strategies, and validates these methods through numerical experiments, advancing the understanding of iterative algorithms in complex game settings.

Contribution

The paper offers the first explicit convergence rate for fictitious play in general mean-field games, including non-potential types, and proposes two novel acceleration techniques.

Findings

Fictitious play converges linearly under certain conditions.

Acceleration strategies improve convergence speed and stability.

Numerical examples confirm theoretical convergence rates.

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.