Well-posedness and mean-field limit estimate of a consensus-based algorithm for multiplayer games
Hui Huang, Jethro Warnett
TL;DR
The paper addresses rigorous well-posedness and quantitative mean-field limits for a derivative-free consensus-based optimization method in nonconvex multiplayer games. It proves existence and uniqueness for both the finite-particle CBO dynamics and the mean-field limit, provides uniform moment bounds, and establishes a explicit rate $N^{-\gamma}$ (with $\gamma$ involving the moment indices and growth exponents) for the mean-field approximation, along with conditions that yield Monte Carlo $\mathcal{O}(N^{-1/2})$ convergence. The analysis relies on Wasserstein stability of the weighted consensus, moment controls, and a Leray–Schauder fixed-point framework, culminating in a Grönwall-based mean-field error bound. These results close gaps in prior work by delivering explicit convergence rates and well-posedness in the multi-species CBO setting, underpinning the theoretical robustness of the consensus-based approach to finding Nash equilibria in multiplayer games.
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.