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Favorable red and black on the integers with a minimum bet

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Kevin Ruth
Kevin Ruth*
Affiliation:
University of Miami
*
∗Postal address: Dept. of Math & Computer Science, College of Arts & Sciences, Coral Gables, FL 33124–4250, USA. Email address: [email protected]
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Abstract

In a superfair red and black gambling house where the player must bet at least 2 units at each stage, a gambler wishes to maximize the probability of reaching a goal integer N before reaching zero. For win probability p > 1/2, when N is even, an optimal strategy is to bet 3 units when you have 3 units or N − 3 units and to bet 2 units otherwise. When N is odd, there are two strategies which are optimal depending on the value of the win probability p. When p is smaller than a certain value, p*, the above strategy is optimal, and when p is larger than this value, the timid strategy is optimal.

Keywords

Superfair red and black gambling housetimid strategyexcessivenessgambler's ruin

MSC classification

Secondary: 90C40: Markov and semi-Markov decision processes 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 837 - 851
Copyright
Copyright © Applied Probability Trust 1999 

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References

Dubins, L. E., and Savage, L. J. (1976). Inequalities for Stochastic Processes: How to Gamble If You Must, 2nd edn. Dover, New York. (First edition 1965.)Google Scholar
Freedman, D. A. (1967). Timid play is optimal. Ann. Math. Statist. 38, 1281–1283.Google Scholar
Heath, D. C., Pruitt, W. E., and Sudderth, W. D. (1971). Subfair red-and-black with a limit. Proc. Amer. Math. Soc. 35, 555–560.Google Scholar
Maitra, A., and Sudderth, W. (1996). Discrete Gambling and Stochastic Games. Springer, New York.Google Scholar
Ross, S. M. (1974). Dynamic programming and gambling models. Adv. App. Prob. 6, 598–606.Google Scholar
Wilkins, J. E. Jr. (1971). The bold strategy in presence of house limit. Proc Amer. Math. Soc. 32, 567–570.Google Scholar