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

Abstract 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.