Abstract

Co-authors: Dan Lanoue (U.C. Berkeley), Justin Salez (Paris 7)

In the Compulsive Gambler process there are $n$ agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$ u_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other’s money. The process seems pedagogically interesting as being intermediate between coalescent-tree models and interacting particle models, and because of the variety of techniques available for its study. Some techniques are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an ``exchangeable over the money elements” property, and a ``token process” construction reminiscent of the Donnelly-Kurtz look-down construction). One can study both kinds of $n o infty$ limit. The process can be defined under weak assumptions on a countable discrete space (nearest-neighbor interaction on trees, or long-range interaction on the $d$-dimensional lattice) and there is also a continuous-space extension called the Metric Coalescent.