We study the following game. Three players start with initial capitals of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$s_{1},s_{2},s_{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> dollars; in each round player <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$P_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> is selected with probability <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </math></EquationSource> </InlineEquation>; then he selects player <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and they play a game in which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$P_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> wins from (resp. loses to) <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> one dollar with probability <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p_{mn}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p_{nm}=1-p_{mn}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mrow> <mi mathvariant="italic">nm</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>p</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>). 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 stationary deterministic strategies.

Three-Gambler Ruin Game: A Game Theoretic Analysis

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2025年4月9日

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Athanasios Kehagias,
Georgios Gkyzis,
Anastasios Karakoulakis,
Aris Kyprianidis

摘要

We study the following game. Three players start with initial capitals of $s_{1},s_{2},s_{3}$ $s_{1}, s_{2}, s_{3}$ dollars; in each round player $P_{m}$ $P_{m}$ is selected with probability $\frac{1}{3}$ $\frac{1}{3}$ ; then he selects player $P_{n}$ $P_{n}$ and they play a game in which $P_{m}$ $P_{m}$ wins from (resp. loses to) $P_{n}$ $P_{n}$ one dollar with probability $p_{mn}$ $p_{mn}$ (resp. $p_{nm}=1-p_{mn}$ $p_{nm} = 1 - p_{mn}$ ). 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 stochastic game and prove that it has at least one Nash equilibrium in stationary deterministic strategies.