Statistical Gambler's Ruin Problem

Abstract

The gambler's ruin problem is one of the most important problems in the emergence of probability. The problem has been long considered “solved” from a probabilistic viewpoint. However, we do not find the solution satisfactory. In this paper, the problem is recast as a statistical problem. Bounds of the estimate are derived over wide classes of priors. Interestingly, the probabilistic estimates ω(1/2) are identified as the most conservative solutions while the plug-in estimates are found to be out of range of the bounds. It implies that, although conservative, the probabilistic estimates ω(1/2) are justified by our analysis while the plug-in estimates are too extreme for estimating the ruin probability of gambler.

Keywords:

• Gambler's ruin problem
• Bayesian robustness analysis
• Beta prior
• AMS 1991 Subject Classification: Primary: 62A15
• 62F35
• Secondary: 62C99

Acknowledgment

The research is supported by NSC89-2118-M-259-013.