From interacting agents to Boltzmann-Gibbs distribution of money

TL;DR

This paper rigorously analyzes a simple model of money exchange among agents, showing that the wealth distribution converges to a geometric distribution over time using entropy methods and propagation of chaos.

Contribution

It provides a rigorous proof of the convergence to the geometric wealth distribution and establishes a quantitative rate of convergence using entropy techniques.

Findings

We prove uniform-in-time propagation of chaos for the model.

The wealth distribution converges exponentially fast to the geometric distribution.

Entropy-entropy dissipation inequalities quantify the convergence rate.

Abstract

We investigate the unbiased model for money exchanges: agents give at random time a dollar to one another (if they have one). Surprisingly, this dynamics eventually leads to a geometric distribution of wealth (shown empirically by Dragulescu and Yakovenko in [11] and rigorously in [2,12,15,18]). We prove a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. This deterministic description is then analyzed by taking advantage of several entropy-entropy dissipation inequalities and we provide a quantitative almost-exponential rate of convergence toward the equilibrium (geometric distribution) in relative entropy.