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A Game Theoretic Analysis of the Three-Gambler Ruin Game

Ath. Kehagias, G. Gkyzis, A. Karakoulakis, A. Kyprianidis

TL;DR

This work analyzes a strategic three-player gambler's ruin variant in which an activated player selects an opponent, with outcomes governed by probabilities $p_{mn}$, and studies the resulting stochastic game to determine equilibria. It proves the existence of at least one Nash equilibrium in deterministic stationary strategies for any $K$ and $p$, by employing a discounted auxiliary game and Markov-chain analysis, and provides an explicit equilibrium characterization for the $K=3$ case. For the general case, it derives payoff equations and Markov-chain structure, shows how to compute absorbing probabilities, and discusses both stationary and nonstationary equilibria, including computational approaches such as MultiValue Iteration and exhaustive enumeration for small $K$. The results illuminate how strategic opponent selection affects winning probabilities in gambler's ruin and establish a foundation for extending to more players, graphs, and alternative payoff criteria, with practical implications for strategic planning in stochastic competitive settings.

Abstract

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.

A Game Theoretic Analysis of the Three-Gambler Ruin Game

TL;DR

This work analyzes a strategic three-player gambler's ruin variant in which an activated player selects an opponent, with outcomes governed by probabilities , and studies the resulting stochastic game to determine equilibria. It proves the existence of at least one Nash equilibrium in deterministic stationary strategies for any and , by employing a discounted auxiliary game and Markov-chain analysis, and provides an explicit equilibrium characterization for the case. For the general case, it derives payoff equations and Markov-chain structure, shows how to compute absorbing probabilities, and discusses both stationary and nonstationary equilibria, including computational approaches such as MultiValue Iteration and exhaustive enumeration for small . The results illuminate how strategic opponent selection affects winning probabilities in gambler's ruin and establish a foundation for extending to more players, graphs, and alternative payoff criteria, with practical implications for strategic planning in stochastic competitive settings.

Abstract

We study the following game. Three players start with initial capitals of dollars; in each round player is selected with probability ; then \emph{he} selects player and they play a game in which wins from (resp. loses to) one dollar with probability (resp. ). 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.
Paper Structure (10 sections, 7 theorems, 61 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Game
  3. The case $K=3$
  4. The General Case
  5. Payoff Equations
  6. Markov Chains
  7. Stationary Equilibria
  8. Nonstationary Equilibria
  9. Computation of Stationary Equilibria
  10. Conclusion

Key Result

Proposition 4.1

If for all $m,n\in \left \{ 1,2,3\right \}$ we have $p_{mn}\in \left( 0,1\right)$ then, for every $\mathbf{x}$, every interior state communicates with all terminal states.

Figures (2)

  • Figure 1: Proportion of succesful MVI runs for 100 randomly selected $\mathbf{p}$ vectors, as a function of $K$.
  • Figure 2: Proportion of succesful MVI runs for 100 randomly selected $\mathbf{p}$ vectors, as a function of $K$.

Theorems & Definitions (7)

  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Proposition 4.5: Fink Fink1961
  • Proposition 4.6
  • Corollary 4.7