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Uniform propagation of chaos for a dollar exchange econophysics model

Roberto Cortez, Fei Cao

TL;DR

This work proves uniform-in-time propagation of chaos for the poor-biased dollar-exchange model in a closed economy of $N$ agents with mean wealth $μ$, validating the mean-field ODE system ${\bf p}'(t)=\mathcal{L}[{f p}(t)]$ and its Poisson equilibrium ${p_n^*}=\frac{μ^n e^{-μ}}{n!}$ for all times. The authors develop a coupling-based framework using Wasserstein distance to connect the $N$-agent dynamics, the multinomial-to-Poisson limit, and the mean-field relaxation to Poisson, delivering explicit $O(N^{-1/2})$ convergence rates. Key steps include a contraction estimate ${\mathcal W}_1(\mathcal{L}(S(t)),\mathcal{L}(R(t))) \le 2 μ e^{-t}$, a coupling proving ${\mathcal W}_1(\mathcal{M}_N, \text{Poisson}(μ)^{\otimes N}) \le \sqrt{2 μ/π}/\sqrt{N}$, and a mean-field convergence bound ${\mathcal W}_1({\bf p}(t),{\bf p}^*) \le 2 μ e^{-t}$. The main result shows that the k-particle marginals approach the tensorized mean-field law uniformly in time, with explicit finite-$N$ rates, thereby justifying the mean-field approximation for long times. The work lays groundwork for extensions to more complex econophysics settings, such as including banks or debt, and suggests avenues to sharpen decay rates via $\chi^2$-based analyses.

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.

Uniform propagation of chaos for a dollar exchange econophysics model

TL;DR

This work proves uniform-in-time propagation of chaos for the poor-biased dollar-exchange model in a closed economy of agents with mean wealth , validating the mean-field ODE system and its Poisson equilibrium for all times. The authors develop a coupling-based framework using Wasserstein distance to connect the -agent dynamics, the multinomial-to-Poisson limit, and the mean-field relaxation to Poisson, delivering explicit convergence rates. Key steps include a contraction estimate , a coupling proving , and a mean-field convergence bound . The main result shows that the k-particle marginals approach the tensorized mean-field law uniformly in time, with explicit finite- rates, thereby justifying the mean-field approximation for long times. The work lays groundwork for extensions to more complex econophysics settings, such as including banks or debt, and suggests avenues to sharpen decay rates via -based analyses.

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.
Paper Structure (7 sections, 5 theorems, 50 equations, 6 figures)

This paper contains 7 sections, 5 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Uniform propagation of chaos
  3. Equilibration of the $N$-agents dynamics
  4. Chaos of the multinomial to Poisson
  5. Equilibration of the mean-field ODE towards Poisson
  6. Proof of the uniform propagation of chaos
  7. Conclusion

Key Result

Proposition 1.1

Let ${\bf p}(t)=\{p_n(t)\}_{n\geq 0}$ be the unique solution of eq:law_limit with $p(0) \in V_\mu$, where is the space of probability mass functions on $\mathbb{N}$ with the pre-fixed mean value $\mu$. Then In particular, we have ${\bf p}(t) \in V_\mu$ for all $t\geq 0$. Moreover, the unique equilibrium distribution ${\bf p}^*=\{p^*_n\}_n$ in $V_\mu$ associated with eq:law_limit is given by the

Figures (6)

  • Figure 1: Left: Illustration of the poor-biased dollar exchange model: at random time, one dollar is passed from a "giver" $i$ to a "receiver" $j$ at a rate proportional to the amount of dollars the "giver" $i$ has. Right: The distribution of wealth for the poor-biased dynamics after $2,000$ unit of time with the average amount of dollar per agent $\mu = 10$, this distribution is well-approximated by a Poisson distribution with mean value $\mu = 10$.
  • Figure 2: Schematic illustration of the limiting ODE system \ref{['eq:law_limit']}.
  • Figure 3: Roadmap for proving convergence results. The approach of taking the large time limit $t\to \infty$ before taking the large population limit $N \to \infty$ is adapted in lanchier_rigorous_2017. An alternative approach is to send $N \to \infty$ first before investigating the large time asymptotic cao_derivation_2021.
  • Figure 4: Schematic illustration of the strategy behind the proof of the uniform in time propagation of chaos for the poor-biased dollar exchange model.
  • Figure 5: Construction of $N$ independent copies of a one-dimensional Poisson process via an infinite collection of i.i.d. exponentially distributed random variables.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Proposition 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 1