A Game Theoretic Analysis of the Three-Gambler Ruin Game

We study the following game. Three players start with initial capitals of s₁,s₂,s₃ dollars; in each round player Pₘ is selected with probability 1/3; then he selects player Pₙ and they play a game in which Pₘ wins from (resp. loses to) Pₙ one dollar with probability pₘₙ (resp. pₙₘ=1-pₘₙ). 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 stochastic game and prove that it has at least one Nash equilibrium in deterministic stationary strategies.