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Persi Diaconis
Persi Diaconis, Stanford University

Gambler’s ruin with three gamblers

Imagine three gamblers with respectively $A$, $B$, $C$ at the start. Each time, a pair of gamblers are chosen (uniformly at random) and a fair coin is flipped. Of course, eventually, one of the gamblers is eliminated and the game continues with the remaining two until one winds up with all $A+B+C$. In poker tournaments (really) it is of interest to know the chances of the six possible elimination orders (e.g. $3,1,2$ means gambler $3$ is eliminated first, then gambler 1, leaving 2 with all the cash). In particular, how do these depend on $A,B,C$? For small $A,B,C$, exact computation is possible, but for $A,B,C$ of practical interest, asymptotics are needed. The math involved is surprising; Whitney and John domains, Carlesson estimates. To test your intuition, recall that if there are two gamblers with $1$ and $N-1$ the chance that the first winds up with all $N$ is $1/N$. With three gamblers with $1,1$ and $N-2$ the chance that the third is eliminated first is $\frac{\operatorname{Const}}{N^3}$. We don't know the answer for four gamblers. This is a report of joint work with Stew Ethier, Kelsey Huston-Edwards and Laurent Saloff-Coste.

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