## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 3 (1980), 451-464.

### Optimal Stopping in an Urn

Wen-chen Chen and Norman Starr

#### Abstract

An urn contains $N$ objects, labelled with the integers $1, \cdots, N$. One object is removed at a time, without replacement. If after $n$ draws the largest number which has been observed is $m_n$, and the process is terminated, we receive a payoff $f(n, m_n)$. For payoff functions $f$ in a certain class, the optimal time to stop is with draw $$\tau_f = \inf\{n \geqslant 0: m_n - n \geqslant j_n\}$$ where the $j_n$ are computable from a simple algorithm, which permits also exact computation of the value $$V_f = E\{f(\tau_f, m_{\tau_f})\}.$$ We also study the behavior of $V_f$ when $N$ is large in special cases.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 3 (1980), 451-464.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994720

**Digital Object Identifier**

doi:10.1214/aop/1176994720

**Mathematical Reviews number (MathSciNet)**

MR573286

**Zentralblatt MATH identifier**

0434.60046

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L15: Optimal stopping [See also 60G40, 91A60]

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Optimal stopping secretary problem dynamic programming urn sampling myopic strategy maximum process

#### Citation

Chen, Wen-chen; Starr, Norman. Optimal Stopping in an Urn. Ann. Probab. 8 (1980), no. 3, 451--464. doi:10.1214/aop/1176994720. https://projecteuclid.org/euclid.aop/1176994720