A Game Theoretic Analysis of the Three-Gambler Ruin Game

We study the following game. Three players start with initial capitals of $s_{1},s_{2},s_{3}$ dollars; in each round player $P_{m}$ is selected with probability $\frac{1}{3}$; then \emph{he} selects player $P_{n}$ and they play a game in which $P_{m}$ wins from (resp. loses to) $P_{n}$ one dollar with probability $p_{mn}$ (resp. $p_{nm}=1-p_{mn}$). When a player loses all his capital he drops out; the game continues until a single player wins by collecting everybody's money. This is a "strategic" version of the classical Gambler's Ruin game. It seems reasonable that a player may improve his winning probability by judicious selection of which opponent to engage in each round. We formulate the situation as a \emph{stochastic game} and prove that it has at least one Nash equilibrium in deterministic stationary strategies.