Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

Pawel Lorek

Abstract:

We give explicit formulas for ruin probabilities in a multidimensional General-
ized 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 gen-
eralizations 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 prob-
lems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied
recently in context of Möbius monotonicity.