  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).
