Limiting distributions of generalized money exchange models

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. This model has been extensively studied in the field of econophysics, particularly by applying ideas from statistical physics, where the money exchange is viewed as analogous to the transfer of energy or particles.

We begin by describing several specific models. The first one is the immediate exchange model proposed in [10]. In this model, the wealth of each agent is represented by a real-valued variable. At each time step, two agents are randomly chosen and give a random fraction of their wealth to each other. The fraction is determined by independent uniformly distributed random variables on the interval $[0,1]$. [10] studied the model through numerical simulations, and later, [11] analytically explored the infinite population version. As a more realistic microscopic model that includes spatial structure in the form of local interactions, [14] formulated the corresponding discrete version as an interacting particle system. We briefly explain their model and result. Consider a finite connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. Each site $x\in\mathcal{V}$ corresponds to an agent, and the economy is represented by the set of agents $\mathcal{V}$. The population size is given by $|\mathcal{V}|=N$. The amount of money each agent holds is represented by the number of coins, where $M_{n}(x)$ represents the number of coins that agent $x\in\mathcal{V}$ possesses at time $n$. The edge set $\mathcal{E}$ represents a social network in which only agents connected by an edge can interact to exchange coins. At each time step, an edge $e=\{x,y\}\in\mathcal{E}$ is chosen uniformly at random from $\mathcal{E}$. Agents $x$ and $y$ independently and uniformly select a random number of their coins to give to each other. Thus, $\{(M_{n}(x))_{x\in\mathcal{V}}\}_{n\geq 0}$ constitutes a time-homogeneous Markov chain. In particular, the total number of coins is conserved in this process, and we denote this total by $L$. [14] proved that the stationary distribution $\mu_{N,L}$ for this Markov chain uniquely exists and gave the following explicit representation.

for every $x\in\mathcal{V}$ and $c\in\{0,1,2,\cdots,L\}$. The first equality is a consequence of the Markov chain convergence theorem. Then, by applying formal calculations to the right-hand side, the authors obtained the following approximation.

for large enough $N$ and $T=\frac{L}{N}$. In this context $T$ represents the average number of coins per agent and is called money temperature by analogy with the temperature in physics and the limit $N\to\infty$ and $T=\frac{L}{N}\to\infty$ are called large population and large money temperature limit. The right-hand side of (1.2) is a probability density function of the Gamma distribution with mean $T$ and shape parameter two, and this is consistent with the predicted result by [10] and [11]. As related results, [4] and [9] examined the duality between the real-valued model and the discrete state version in a continuous time setting. In addition, [9] considered the case where the exchange fraction is determined by a Beta distribution.

We also present two models with different exchange rules that we address in this paper.

So far, [14] has formulated several money exchange models as Markov chains with spatial structure, characterized their stationary distributions and further obtained their formal approximations. The conclusions explain the distribution of wealth in a sense, as predicted by numerical simulations and other methods. However, the approximation $``\approx"$ in (1.2) is not mathematically valid. The left-hand side is defined only for non-negative integers $c$, while the right-hand represents a probability density function on $\mathbb{R}_{+}$. The derivation of the approximation (1.2), although merely a formal calculation, seems to depend on assumptions that are not fully clarified. Specifically, in the proofs of Theorems 1, 2 and 3 in [14], $L+1,L+2,\cdots,L+N$ are replaced with $L$ for sufficiently large $L$ and $N$ satisfying $N\ll L$. On the other hand, $L-c+1,L-c+2,\cdots,L-c+N$ are replaced with $L-c$, rather than $L$, even when $c\ll L$. Additionally, $c+1$ has been conveniently replaced with $c$. As a matter of fact, it seems unnatural to consider the approximation of $\mu_{N,L}(\{\xi\in\mathbb{Z}_{+}^{\mathcal{V}};\xi(x)=c\})$ for each $c$ since the average number of coins per agent diverges in the limit $T=\frac{L}{N}\to\infty$. The money exchange model describes the microscopic movement of money, however, our goal is to derive the macroscopic distribution of wealth in the limit $N\to\infty$ and $\frac{L}{N}\to\infty$. In order to do that, we should analyze the convergence of $\mu_{N,L}$ under appropriate scaling. Therefore, the primary objective of this paper is to provide a mathematically rigorous justification of the approximation (1.2) and to clearly demonstrate the convergence of the wealth distribution. Furthermore, we generalize the rules of money exchange as follows:

• •

  Randomly select the number of coins to pass or save based on a probability distribution that depends on the number of coins, rather than using the uniform distribution.

• •

  Allow the exchange or redistribution of coins among randomly selected groups of three or more agents.

These generalizations appear natural from both mathematical and economic perspectives. We note that as a generalization of the exchange rules in the immediate exchange model, [23] considered a broader class of models where mass is split, exchanged and merged. In [13], [15] and [16], the authors formulated other money exchange models as Markov chains and studied their stationary distributions and formal approximations in a manner similar to (1.1) and (1.2). Also, the mixing time has been studied recently for the binomial splitting model and the symmetric beta-binomial splitting model, which are variations of the uniform reshuffling model (cf. [21], [22]).

Before introducing our models and results we prepare several notations. In the following, $\mathbb{Z}_{+}=\{0,1,2,\cdots\}$ denotes the set of non-negative integers and $\mathbb{N}=\{1,2,3,\cdots\}$ denotes the set of positive integers. $[a]$ denotes the integral part of $a>0$ and we set $a\vee b:=\max\{a,b\}$ and $a\wedge b:=\min\{a,b\}$ for $a,b\in\mathbb{R}$. For a finite set $A$, $|A|$ denotes its cardinality. For two sequences of positive numbers $\{a_{n}\}$ and $\{b_{n}\}$, $a_{n}\sim b_{n}$ means that $\lim\limits_{n\to\infty}\frac{a_{n}}{b_{n}}=1$. A function $f$ on $\mathbb{R}^{\mathbb{Z}}$ is called local if it depends only on finitely many coordinates. For a probability measure $\mu$, $E^{\mu}[\,\cdot\,]$ denotes the expectation with respect to $\mu$.

Let us state our model precisely. We adopt the standard notations commonly used in the study of interacting particle systems. For $A\subset\mathbb{Z}$ and $L\in\mathbb{N}$, we define the configuration space

When $A=\Lambda_{N}:=\{1,2,\cdots,N\}$, $\Omega(A,L)$ is denoted by $\Omega_{N}(L)$. Consider now an economy populated by many agents. Each site $x\in\Lambda_{N}$ corresponds to an agent and we assume that the economy can be represented by the set of agents $\Lambda_{N}$. $N$ corresponds to the population size. For each $\eta\in\Omega_{N}(L)$, we interpret $\eta(x)$, $x\in\Lambda_{N}$ not as the number of particles, but as the number of coins held by agent $x$. $\rho$ denotes a probability distribution on $\mathcal{D}_{N}=\{A\subset\Lambda_{N};|A|\geq 2\}$, namely $\rho(A)\geq 0$ for every $A\subset\Lambda_{N}$ with $|A|\geq 2$ and $\sum\limits_{A\in\mathcal{D}_{N}}\rho(A)=1$. This represents the distribution that determines which agent handles the money exchange at each time step. We also take a non-negative function $g(\not\equiv 0)$ defined on $\mathbb{Z}_{+}$. Now, we introduce three money exchange models.

As the first result, we characterize the stationary distributions of these Markov chains.

Now, we are in the position to state the main result of this paper. To justify the limit $N\to\infty$ and $\frac{L}{N}\to\infty$, we assume that the total number of coins $L=L_{N}$ satisfies $\lim\limits_{N\to\infty}\frac{L_{N}}{Na_{N}}=T$ for some $T>0$ and divergent sequence $\{a_{N}\}_{N\geq 1}$. Then, we can prove that the law of the scaled field $\bigl{\{}\frac{1}{a_{N}}\eta(x)\bigr{\}}_{x\in\Lambda_{N}}$ under $\mu_{N,L}$ or $\pi_{N,L}$ converges to the i.i.d. product of probability distributions on $\mathbb{R}_{+}$. Its marginal distribution depends on the asymptotic behavior of the weight function $g$.

As an easy consequence of Proposition 1.1 and Theorem 1.1, we obtain the following.

In the rest of the paper we provide the proofs of Proposition 1.1 and Theorem 1.1 in Sections 2 and 3, respectively. We give some comments about the strategy of the proof. The proof of Proposition 1.1 is standard. It is not difficult to see that our models are irreducible aperiodic Markov chains on the finite state space $\Omega_{N}(L)$. All we have to do is to characterize the unique stationary distribution, which is achieved by carefully verifying the detailed balance condition. Namely, we demonstrate that our models are reversible Markov chains, and the reversible distributions for $\{X_{n}\}_{n\geq 0}$ and $\{Y_{n}\}_{n\geq 0}$ are given by (1.4). With respect to the proof of Theorem 1.1, the convergence of the marginal distribution of $\mu_{N,L}$ is closely related to the equivalence of ensembles (cf. [8], [12]). The sequence $\{G(k)\}_{k\in\mathbb{Z}_{+}}$ is neither a probability distribution nor generally summable over $k$. However, multiplying $G(k)$ by the exponential factor $s^{k}$, $s\in[0,1)$ ensures the convergence of $\sum\limits_{k\in\mathbb{Z}_{+}}s^{k}G(k)$ under the assumption on $g$. The stationary distribution (1.4) can then be interpreted as the microcanonical distribution of an i.i.d. product, where the one-site marginal distribution is of exponential type and proportional to $G$. Instead of the usual condition $\frac{1}{N}\sum\limits_{x\in\Lambda_{N}}\eta(x)\to m\in(0,\infty)$ as $N\to\infty$, we consider the condition $\frac{1}{Na_{N}}\sum\limits_{x\in\Lambda_{N}}\eta(x)\to T\in(0,\infty)$ and investigate the convergence of the law of the scaled field $\frac{1}{a_{N}}\eta$ under the corresponding microcanonical distribution. We adapt the proof of the equivalence of ensembles from [12, Appendix 2] to this unusual setting. In particular, the local limit theorem for a triangular array of random variables plays an important role in the argument.

Throughout the paper $C$, $C^{\prime}$, $C^{\prime\prime}$ represent positive constants that do not depend on the size of the system $N$, but may depend on other parameters. These constants in various estimates may change from place to place in the paper.