From Gini index as a Lyapunov functional to convergence in Wasserstein distance

TL;DR

This paper investigates how the Gini index, used as a Lyapunov functional in infinite-dimensional ODE systems modeling probability distributions, relates to convergence in Wasserstein distance and other metrics, clarifying the link between index convergence and distribution convergence.

Contribution

The paper establishes theoretical results connecting Gini index convergence to Wasserstein and other metric convergences in the context of mean-field models.

Findings

Gini index convergence implies Wasserstein convergence under certain conditions

Relationships between Gini index and $ ext{L}^p$ distances are characterized

Results clarify when index convergence guarantees distribution convergence

Abstract

In several recent works on infinite-dimensional systems of ODEs \cite{cao_derivation_2021,cao_explicit_2021,cao_iterative_2024,cao_sticky_2024}, which arise from the mean-field limit of agent-based models in economics and social sciences and model the evolution of probability distributions (on the set of non-negative integers), it is often shown that the Gini index serves as a natural Lyapunov functional along the solution to a given system. Furthermore, the Gini index converges to that of the equilibrium distribution. However, it is not immediately clear whether this convergence at the level of the Gini index implies convergence in the sense of probability distributions or even stronger notions of convergence. In this paper, we prove several results in this direction, highlighting the interplay between the Gini index and other popular metrics, such as the Wasserstein distance and the usual $\ell^p$ distance, which are used to quantify the closeness of probability distributions.