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

Paper: arXiv 2409.15225

Authors: Fei Cao

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs establishing theoretical bounds linking the Gini index to Wasserstein distance and standard distance metrics in mean-field ODE systems. However, it is entirely theoretical without empirical backtesting, financial market data validation, or algorithmic implementations.

Research Flowchart

  flowchart TD
    A["Research Question:<br/>Does Gini index convergence imply<br/>Wasserstein/ℓᵖ convergence?"] --> B["Input Data:<br/>Probability distributions on ℕ₀<br/>from Agent-Based Mean-Field ODEs"]
    B --> C["Methodology:<br/>Analyze Gini index as a<br/>Lyapunov functional"]
    C --> D["Mathematical Derivations:<br/>Relate Gini Index to<br/>Wasserstein & ℓᵖ Metrics"]
    D --> E["Key Outcome 1:<br/>Gini convergence bounds<br/>Wasserstein distance"]
    D --> F["Key Outcome 2:<br/>Established conditions for<br/>stronger ℓᵖ distribution convergence"]