A mathematical model for a gaming community

TL;DR

This paper models a gaming community where individuals repeatedly bet on coin tosses, showing that asset distribution converges to an exponential distribution, indicating a stable equilibrium in such zero-sum games.

Contribution

It introduces a mathematical model demonstrating that asset distribution in a betting community converges to an exponential distribution, regardless of bet size, under no-bankruptcy constraints.

Findings

Asset distribution converges to exponential distribution.

Exponential distribution is a stable fixed point.

Numerical experiments confirm analytical results.

Abstract

We consider a large community of individuals who mix strongly and meet in pairs to bet on a coin toss. We investigate the asset distribution of the players involved in this zero-sum repeated game. Our main result is that the asset distribution converges to the exponential distribution, irrespective of the size of the bet, as long as players can never go bankrupt. Analytical results suggests that the exponential distribution is a stable fixed point for this zero-sum repreated game. This is confirmed in numerical experiments.