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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
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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Keywords: Bold strategy, gambling, optimal strategy
Article copyright: © Copyright 1972 American Mathematical Society

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