## The Annals of Statistics

- Ann. Statist.
- Volume 5, Number 4 (1977), 734-745.

### Discounted and Rapid Subfair Red-and-Black

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 5, Number 4 (1977), 734-745.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343896

**Digital Object Identifier**

doi:10.1214/aos/1176343896

**Mathematical Reviews number (MathSciNet)**

MR438478

**Zentralblatt MATH identifier**

0379.60044

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Secondary: 93E99: None of the above, but in this section

**Keywords**

Gambling problem red-and-black optimal strategy bold strategy stop rule

#### Citation

Klugman, Stuart. Discounted and Rapid Subfair Red-and-Black. Ann. Statist. 5 (1977), no. 4, 734--745. doi:10.1214/aos/1176343896. https://projecteuclid.org/euclid.aos/1176343896