Analytic treatment of a trading market model

TL;DR

This paper provides an analytical and numerical study of a simple trading market model with two agents, revealing different universality classes for zero and finite savings scenarios and validating results with existing numerical methods.

Contribution

It introduces an integral nonlinear equation for the money distribution in a two-agent market model and analyzes both zero and finite savings cases, including an analytical solution for zero savings.

Findings

Zero savings case solvable analytically.

Finite savings case solved numerically.

Results agree with previous numerical studies.

Abstract

We mathematically analyze a simple market model where trading at each point in time involves only two agents with the sum of their money being conserved and with neither parties resulting with negative money after the interaction process. The exchange involves random re-distribution among the two players of a fixed fraction of their total money. We obtain a simple integral nonlinear equation for the money distribution. We find that the zero savings and finite savings cases belong to different universality classes. While the zero savings case can be solved analytically, the finite savings solution is obtained by numerically solving the integral equation. We find remarkable agreement with results obtained by other researchers using sophisticated numerical techniques.