The 4-player gambler's ruin problem

TL;DR

This paper estimates the probability that a dominant player loses first in a 4-player gambler's ruin game, using spectral analysis of spherical Laplacian eigenvalues and numerical methods.

Contribution

It introduces a novel approach linking gambler's ruin probabilities to Dirichlet eigenvalues of the spherical Laplacian, providing explicit formulas and numerical estimates.

Findings

Probability of dominant player losing first is approximately N^{-5.68}

Derived explicit relation between probability decay rate and Laplacian eigenvalue

Numerical estimation of eigenvalues using finite-difference algorithm

Abstract

This work explains how to utilize earlier results by P. Diaconis, K. Houston-Edwards and the second author to estimate probabilities related to the 4-player gambler ruin problem. For instance, we show that the probability that a very dominant player (i.e., a player starting with all but 3 chips distributed among the remaining players) is first to loose is of order $N^{-\alpha}$ where $\alpha$ is approximately $5.68$. In the $3$-player game, this probability is or order $N^{-3}$. We note it is futile to attempt to give heuristic/intuitive explanations for the value of $\alpha$. This value is obtained via an explicit formula relating $\alpha$ to the Dirichlet eigenvalue $\lambda$ (zero boundary condition) of the spherical Laplacian in the equilateral spherical triangle on the unit sphere $\mathbb S^2$ that corresponds to a unit simplex with one vertex placed at the origin in Euclidean $3$-space. The value of $\lambda$ is estimated using a finite-difference-type algorithm developed by Grady Wright.