From the binomial reshuffling model to Poisson distribution of money

TL;DR

This paper introduces a new wealth redistribution model based on binomial reshuffling, demonstrating that wealth distribution converges to a Poisson distribution as the population grows, supported by theoretical proofs and numerical simulations.

Contribution

It presents a novel binomial reshuffling model and proves convergence to a Poisson distribution in the mean-field limit, linking stochastic dynamics to deterministic equations.

Findings

Wealth distribution converges to Poisson in the large population limit.

Numerical simulations support the theoretical convergence and suggest possible improvements.

The model extends the uniform reshuffling framework in econophysics.

Abstract

We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $X_i$ and $X_j$ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $B \circ (X_i+X_j)$. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics [2,14]. As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. The main result of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $2$-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.