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

TL;DR

This paper analyzes a modified asset exchange model with dishonest agents called probabilistic cheaters, using mean-field theory to describe the large population limit and discovering a mixture of distributions and a new entropy functional.

Contribution

It introduces a mean-field framework for a variant of the BDY model with probabilistic cheaters and proves convergence to a stationary mixture distribution.

Findings

Convergence of the mean-field system to a stationary mixture distribution.

Introduction of a generalized entropy functional for the model.

Identification of a novel mixture of geometric distributions in the stationary state.

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.