Well-posedness and mean-field limit estimate of a consensus-based algorithm for multiplayer games

TL;DR

This paper provides theoretical guarantees for a consensus-based algorithm in multiplayer games, including well-posedness and quantitative mean-field limit estimates, enhancing understanding of its convergence and stability.

Contribution

It establishes well-posedness and derives a quantitative mean-field limit estimate, filling theoretical gaps in previous work on consensus-based Nash equilibrium algorithms.

Findings

Proves well-posedness of finite particle and mean-field models

Provides a quantitative estimate of the mean-field limit

Addresses theoretical gaps in convergence analysis

Abstract

Recently, the paper [12] introduces a derivative-free consensus-based particle method that finds the Nash equilibrium of non-convex multiplayer games, where it proves the global exponential convergence in the sense of mean-field law. This paper aims to address theoretical gaps in [12], specifically by providing a quantitative estimate of the mean-field limit with respect to the number of particles, as well as establishing the well-posedness of both the finite particle model and the corresponding mean-field dynamics.