A Game Theoretic Analysis of the Three-Gambler Ruin Game
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911914231070720 |
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| author | Kehagias, Ath. Gkyzis, G. Karakoulakis, A. Kyprianidis, A. |
| author_facet | Kehagias, Ath. Gkyzis, G. Karakoulakis, A. Kyprianidis, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Game Theoretic Analysis of the Three-Gambler Ruin Game Kehagias, Ath. Gkyzis, G. Karakoulakis, A. Kyprianidis, A. Computer Science and Game Theory Probability 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. |
| title | A Game Theoretic Analysis of the Three-Gambler Ruin Game |
| topic | Computer Science and Game Theory Probability |
| url | https://arxiv.org/abs/2406.07878 |