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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Subfair red-and-black with a limit


Authors: David C. Heath, William E. Pruitt and William D. Sudderth
Journal: Proc. Amer. Math. Soc. 35 (1972), 555-560
MSC: Primary 90D99
DOI: https://doi.org/10.1090/S0002-9939-1972-0309574-5
MathSciNet review: 0309574
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Abstract: Suppose a gambler has an initial fortune in (0,1) and wishes to reach 1. It is known that, for a subfair red-and-black casino, the optimal strategy is always to bet $ \min (f,1 - f)$ whenever the gambler's current fortune is f. Furthermore, the gambler should likewise play boldly if there is a house limit z which is the reciprocal of a positive integer; i.e., he should bet $ \min (f,1 - f,z)$. We show that if $ 1/(n + 1) < z < 1/n$ for some integer $ n \geqq 3$ or if z is irrational and $ \frac{1}{3} < z < \frac{1}{2}$, then bold play is not necessarily optimal.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0309574-5
Keywords: Bold strategy, gambling, optimal strategy
Article copyright: © Copyright 1972 American Mathematical Society