The Compulsive Gambler Process

TL;DR

The paper introduces the compulsive gambler process, analyzing its fundamental properties, including martingale structure and exchangeability, and explores extensions such as sparse graph interactions and a continuous-space version called the metric coalescent.

Contribution

It formally defines the compulsive gambler process, investigates its properties, and discusses extensions like sparse graph interactions and the metric coalescent.

Findings

Martingale structure of the process

Comparison with Kingman coalescent

Construction similar to Donnelly-Kurtz look-down

Abstract

In the compulsive gambler process there is a finite set of agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money. We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an "exchangeable over the money elements" property, and a construction reminiscent of the Donnelly-Kurtz look-down construction). Several directions for possible future research are described. One -- where agents meet neighbors in a sparse graph -- is studied here, and another -- a continuous-space extension called the {\em metric coalescent} -- is studied in Lanoue (2014).