Rigorous proof of the Boltzmann-Gibbs distribution of money on connected graphs

TL;DR

This paper rigorously proves that in models of economic agents exchanging money, the distribution converges to an exponential (Boltzmann-Gibbs) form on connected graphs, and also explores a related model with Poissonian distribution.

Contribution

It provides the first rigorous proof of the Boltzmann-Gibbs distribution for money in agent-based models on connected graphs, extending previous numerical findings.

Findings

Distribution converges to exponential on connected graphs

In a related model, the distribution is Poissonian

Universal convergence regardless of graph structure

Abstract

Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann-Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution is universal in the sense that it holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.