Conditional Gambler’s Ruin Problem with Arbitrary Winning and Losing Probabilities with Applications

• Published in: Methodology and computing in applied probability Vol. 27; no. 3; p. 54
• Main Authors: Lorek, Paweł, Markowski, Piotr
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2025; Springer Nature B.V
• Subjects: Absorption; Business and Management; Economics; Eigenvalues; Electrical Engineering; Life Sciences; Mathematics and Statistics; Probability; Random variables; Random walk; Random walk theory; Statistics; 60J10; Strong stationary dual chain; Birth and death chain; Conditional absorption time; Random walk on a polygon; Möbius monotonicity; Random walk on a circle; 60G40; Gambler’s ruin problem; 60J80
• DOI: 10.1007/s11009-025-10181-7
• ISSN: 1387-5841

In this paper we provide formulas for the expectation of a conditional game duration in a finite state-space one-dimensional gambler’s ruin problem with arbitrary winning p ( n ) and losing q ( n ) probabilities ( i.e.,  they depend on the current fortune). The formulas are stated in terms of the parameters of the system. Beyer and Waterman ( Math Mag , 50 (1):42–45, 1977) showed that for the classical gambler’s ruin problem the distribution of a conditional absorption time is symmetric in p and q . Our formulas imply that for non-constant winning/losing probabilities the expectation of a conditional game duration is symmetric in these probabilities ( i.e.,  it is the same if we exchange p ( n ) with q ( n )) as long as a ratio q ( n )/ p ( n ) is constant. Most of the formulas are applied to a non-symmetric random walk on a circle/polygon. Moreover, for a symmetric random walk on a circle we construct an optimal strong stationary dual chain – which turns out to be an absorbing, non-symmetric, birth and death chain. We apply our results and provide a formula for its expected absorption time, which is the fastest strong stationary time for the aforementioned symmetric random walk on a circle. This way we improve upon a result of Diaconis and Fill ( Ann Prob , 18 (4):1483–1522, 1990), where strong stationary time – however not the fastest – was constructed. Expectations of the fastest strong stationary time and the one constructed by Diaconis and Fill differ by 3/4, independently of a circle’s size.