Mean-field analysis of a random asset exchange model with probabilistic cheaters

TL;DR

This work analyzes a two-type variant of the Bennati-Dragulescu-Yakovenko (BDY) money-exchange model with honest players and probabilistic cheaters. In the mean-field limit, the evolution is governed by coupled nonlinear ODEs for ${\bf p}^h(t)$ and ${\bf p}^c(t)$ under a total wealth constraint, yielding a stationary mixture of geometric distributions. The authors prove convergence to the equilibrium pair (a geometric distribution for each subpopulation) by establishing a generalized entropy functional $\mathrm{H}[({\bf p}^c, {\bf p}^h)]$ that increases along trajectories, effectively an H-theorem for this system, and they derive a variational characterization of the equilibrium as a maximum-entropy state subject to the mean wealth constraint. They also perform a linear stability analysis and study how the cheating probability $\gamma$ influences wealth inequality, showing that the Gini index of the equilibrium increases with $\gamma$. The results introduce a novel mean-field framework with a convex mixture of geometric laws and a generalized entropy, with implications for understanding dishonest behavior and wealth disparities in simplified econophysics models.

Abstract

We investigate a variant of the standard Bennati-Dragulescu-Yakovenko (BDY) game \cite{dragulescu_statistical_2000} inspired by the very recent work \cite{blom_hallmarks_2024}, where agents involving in a money exchange dynamics are classified into two distinct types which are termed as probabilistic cheaters and honest players, respectively. A probabilistic cheater has a positive probability of declaring to have no money to give to other agents in the system, resulting in a potential financial benefits from being dishonest about his/her financial status. We provide a mean-field description of the agent-based model (in terms of a coupled infinite dimensional system of nonlinear ODEs), in the large population limit where the number of players is sent to infinity, and proves convergence of the coupled mean-field system to its stationary distribution (provided by a mixture of geometric distributions). In particular, the model gives rise to a novel formulation involving a mixture of probability distributions, thereby motivating the introduction of a unusual (generalized) entropy functional tailored to the associated mean-field system.

Figures (6)

• Figure 1: Illustration of a variant of the BDY game involving probabilistic cheaters: at random times, a "giver" $i$ picked uniformly at random is selected to give one dollar to a "receiver" $j$ chosen uniformly at random as well. If agent $i$ has no dollar (i.e., $S_i = 0$) then nothing happens. Otherwise (i.e., $S_i\geq 1$), agent $i$ will give one dollar to agent $j$ if agent $i$ is an honest player (or equivalently $i \in \mathcal{H}$), and agent $i$ will give one dollar to agent $j$ with probability $1-\gamma$ if agent $i$ is a probabilistic cheater (i.e., $i \in \mathcal{C}$).
• Figure 2: Left: Distribution of money for the agent-based model with $N = 2,000$ agents after $20,000$ units of time, using $\mu =5$, $n_c = n_h =0.5$, $\gamma = 0.5$ and the initial condition $S_i(0) = \mu$ for all $1\leq i \leq N$. Right: Simulation of the coupled mean-field ODE systems \ref{['eq:law_limit_honest']} and \ref{['eq:law_limit_cheater']} for $0\leq t\leq 500$, using $\mu =5$, $n_c = n_h =0.5$, $\gamma = 0.5$ and the initial condition ${\bf p}^h(0) = {\bf p}^c(0) = \delta_\mu$, where $\delta_\mu \in \mathcal{P}(\mathbb N)$ is Dirac-type distribution whose only non-zero component is located at its $(\mu+1)$-th coordinate. We observe that in both scenarios the terminal distributions are well-approximated by a convex combination of geometric distributions given by \ref{['eq:equilibria']} below.
• Figure 3: Distribution of money for the agent-based model with $N = 2,000$ agents after $20,000$ units of time, using $\mu =5$, $n_c = n_h =0.5$, $\gamma = 0.5$ and the initial condition $S_i(0) = \mu$ for all $1\leq i \leq N$. Left: Wealth distribution among probabilistic cheaters. Right: Wealth distribution among honest players. We emphasize that for both sub-systems the large-time distributions are close to a geometric distribution given by ${\bf p}^c$ and ${\bf p}^h$\ref{['eq:equilibria']}, respectively.
• Figure 4: Left: Evolution of the numerical solution ${\bf p}^c(t)$ of the ODE system \ref{['eq:law_limit_cheater']} at various times. Right: Evolution of the numerical solution ${\bf p}^h(t)$ of the ODE system \ref{['eq:law_limit_honest']} at various times. For both sub-systems, ${\bf p}^c(t=500)$ and ${\bf p}^h(t=500)$ are almost indistinguishable from their respective geometric equilibrium distributions $\bar{{\bf p}}^c$ and $\bar{{\bf p}}^h$.
• Figure 5: Left: Evolution of the entropy-like $\mathrm{H}$ functional $\mathrm{H}[({\bf p}^c, {\bf p}^h)]$ over $0\leq t \leq 500$. Right: Evolution of $\mathrm{H}[(\bar{{\bf p}}^c, \bar{{\bf p}}^h)] - \mathrm{H}[({\bf p}^c, {\bf p}^h)]$ over $0\leq t \leq 500$.

Theorems & Definitions (8)

• Lemma 2.1
• Proposition 2.2
• Lemma 3.1: Variational characterization of the geometric pair $(\bar{{\bf p}}^c, \bar{{\bf p}}^h)$
• Theorem 1: Entropy production and convergence to equilibrium
• Lemma 3.2
• Proposition 3.3
• Definition 1: Gini index
• Proposition 4.1