A symbolic computational approach to the generalized gambler's ruin problem in one and two dimensions

TL;DR

This paper demonstrates how symbolic computation can be used to analyze the generalized gambler's ruin problem in one and two dimensions, introducing a new mirror step and providing closed-form formulas for probability and duration.

Contribution

It introduces a symbolic computational approach to the gambler's ruin problem and proposes a new generalization with a mirror step, offering closed-form solutions.

Findings

Efficient algorithms for generalized gambler's ruin in 1D and 2D

Closed formulas for probability and expected duration with mirror step

Extension of classical gambler's ruin with new step type

Abstract

The power of symbolic computation, as opposed to mere numerical computation, is illustrated with efficient algorithms for studying the generalized gambler's ruin problem in one and two dimensions. We also consider a new generalization of the classical gambler's ruin where we add a third step which we call the mirror step. In this scenario, we provide closed formulas for the probability and expected duration.