Uniform propagation of chaos for a dollar exchange econophysics model

TL;DR

This paper proves that as the number of agents in a money exchange model increases, their collective behavior converges uniformly over time to a deterministic mean-field limit, validating the use of simplified equations.

Contribution

It establishes a uniform-in-time propagation of chaos result for the model, providing rigorous justification for the mean-field approximation in the econophysics context.

Findings

Convergence of the money distribution to a Poisson distribution as agents grow large.

Validation of mean-field ODEs as accurate approximations for large systems.

Uniform-in-time propagation of chaos proven for the model.

Abstract

We study the poor-biased model for money exchange introduced in [2]: agents are being randomly picked at a rate proportional to their current wealth, and then the selected agent gives a dollar to another agent picked uniformly at random. Simulations of a stochastic system of finitely many agents as well as a rigorous analysis carried out in [2,16] suggest that, when both the number of agents and time become large enough, the distribution of money among the agents converges to a Poisson distribution. In this manuscript, we establish a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which justifies the validity of the mean-field deterministic infinite system of ordinary differential equations as an approximation of the underlying stochastic agent-based dynamics.