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

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.