A Game Theoretic Analysis of the Three-Gambler Ruin Game
<|reference_start|>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.<|reference_end|>
@article{kehagias2024a,
    title={A Game Theoretic Analysis of the Three-Gambler Ruin Game},
    author={Ath. Kehagias, G. Gkyzis, A. Karakoulakis, A. Kyprianidis},
    journal={arXiv preprint arXiv:2406.07878},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.07878},
    primaryClass={cs.GT math.PR}
}