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July, 1977 Discounted and Rapid Subfair Red-and-Black
Stuart Klugman
Ann. Statist. 5(4): 734-745 (July, 1977). DOI: 10.1214/aos/1176343896


A gambler seeks to maximize the expected utility earned upon reaching a goal in a game where he is allowed at each stage to stake any amount of his current fortune. He wins each bet with probability $w$. In the discounted case the utility at the goal is $\beta^n$ where $\beta$, the discount factor, is in $(0, 1)$ and $n$ is the number of plays used to reach the goal. In the rapid case the utility at the goal is 1 and the gambler seeks to minimize his expected playing time given he reaches the goal. Here all optimal strategies are characterized when $w \leqq \frac{1}{2}$ for the discounted case and when $w < \frac{1}{2}$ for the rapid case. It is shown that when $w < \frac{1}{2}$ the set of rapidly optimal strategies coincides with the set of optimal strategies for the discounted case.


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Stuart Klugman. "Discounted and Rapid Subfair Red-and-Black." Ann. Statist. 5 (4) 734 - 745, July, 1977.


Published: July, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0379.60044
MathSciNet: MR438478
Digital Object Identifier: 10.1214/aos/1176343896

Primary: 60G35
Secondary: 93E99

Keywords: bold strategy , Gambling problem , optimal strategy , red-and-black , stop rule

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 4 • July, 1977
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