Generalized Gambler's Ruin Problem: explicit formulas via Siegumd duality

TL;DR

This paper derives explicit formulas for ruin probabilities in a multidimensional generalized gambler's ruin game involving multiple players with arbitrary win/loss probabilities, using Markov chain dualities.

Contribution

It introduces a novel approach using Siegmund duality to solve complex ruin problems, extending previous models with a unified explicit formula framework.

Findings

Explicit ruin probability formulas for multidimensional games

Application of Siegmund duality in Markov chains for ruin problems

Generalization of previous gambler's ruin models

Abstract

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler's ruin problem. The generalization is best interpreted as a game of one player against $d$ other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of M\"obius monotonicity.