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Limiting distributions of generalized money exchange models

Hironobu Sakagawa

TL;DR

This work provides a rigorous foundation for generalized money-exchange models in econophysics by formulating discrete-time interacting particle systems for immediate exchange, random saving, and uniform reshuffling. It derives the exact stationary distributions and proves that, under a large-population and large-money-temperature scaling with $ rac{L_N}{N a_N} o T$ and $a_N o\

Abstract

The "Money Exchange Model" is a type of agent-based simulation model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model and the uniform saving model both of which are types of money exchange models, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that under appropriate scaling, the asymptotic wealth distribution converges to a Gamma distribution or an exponential distribution for both models. The limiting distribution depends on the weight function that affects the probability distribution of the number of coins exchanged by each agent. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.

Limiting distributions of generalized money exchange models

TL;DR

This work provides a rigorous foundation for generalized money-exchange models in econophysics by formulating discrete-time interacting particle systems for immediate exchange, random saving, and uniform reshuffling. It derives the exact stationary distributions and proves that, under a large-population and large-money-temperature scaling with and $a_N o\

Abstract

The "Money Exchange Model" is a type of agent-based simulation model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model and the uniform saving model both of which are types of money exchange models, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that under appropriate scaling, the asymptotic wealth distribution converges to a Gamma distribution or an exponential distribution for both models. The limiting distribution depends on the weight function that affects the probability distribution of the number of coins exchanged by each agent. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.
Paper Structure (5 sections, 6 theorems, 49 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Money exchange models
  3. Model description and results
  4. Proof of Proposition \ref{['prop1']}
  5. Proof of Theorem \ref{['thm1']}

Key Result

Proposition 1.1

Let $N, L \in \mathbb{N}$ be fixed. We assume that the hypergraph $(\Lambda_N, \mathcal{D}_{N, \rho})$ is connected where the hyperedge set $\mathcal{D}_{N, \rho}$ is defined by $\mathcal{D}_{N, \rho} = \{A\subset \Lambda_N; |A|\geq2, \rho(A)>0\}$.

Figures (2)

  • Figure 1: Simulation results for a single realization of the immediate exchange model and the random saving model with $g(k)=(k+1)^{\alpha}$ where $\alpha = 1, 3$. The number of agents is $N=10^4$ and the total number of coins is $L=10^6$, namely the average number of coins per agent equals to $100$. The initial condition is set to a constant configuration $X_0 \equiv 100$ or $Y_0 \equiv 100$. $\rho$ is distributed uniformly over the edge set $\{\{x, y\}; x, y\in \Lambda_N, x\ne y\}$, and in the immediate exchange model, swapping shall always be performed between the selected edges. The gray histograms represent the wealth distribution, i.e. the proportion of agents holding a specific number of coins after $n=10^5$ updates. The dotted line is the graph of the probability density function of the Gamma distribution: $f_{a, b}(r)= \frac{1}{\Gamma(a)b^{a}} r^{ a-1} e^{-\frac{1}{b}r}$ with the shape parameter $a=\alpha+2$ and the scale parameter $b= \frac{100}{\alpha+2}$.
  • Figure 2: Simulation results for a single realization of the immediate exchange model and the random saving model with $g(k)=(k+1)^{\alpha}$ where $\alpha = -1, -2$. The settings of the simulations are the same as Figure 1. The dotted line is the graph of the probability density function of the exponential distribution: $f_{\lambda}(r)= {\lambda} e^{-\lambda r}$ with the parameter $\lambda=\frac{1}{100}$. When the weight function $g$ decays rapidly in the immediate exchange model, the probability of each agent exchanging only a small number of coins at a time increases, leading to a longer convergence time to reach a steady state. The simulation result for the case $g(k)=(k+1)^{-2}$ reflects this and $n=10^5$ updates are not sufficient for convergence (the middle left). However, after $n=10^6$ updates, the histograms approach the limiting probability density function (the lower left). Also, the result for the case $N=10^3$ and $n=10^6$ indicates that we need to take a large population $N$ (and hence a large $L$) for convergence (the lower right).

Theorems & Definitions (12)

  • Proposition 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • ...and 2 more