## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 1 (1991), 342-353.

### Minimax-Optimal Stop Rules and Distributions in Secretary Problems

Theodore P. Hill and Ulrich Krengel

#### Abstract

For the secretary (or best-choice) problem with an unknown number $N$ of objects, minimax-optimal stop rules and (worst-case) distributions are derived, under the assumption that $N$ is a random variable with unknown distribution, but known upper bound $n$. Asymptotically, the probability of selecting the best object in this situation is of order of $(\log n)^{-1}$. For example, even if the only information available is that there are somewhere between 1 and 100 objects, there is still a strategy which will select the best item about one time in five.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 1 (1991), 342-353.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990548

**Digital Object Identifier**

doi:10.1214/aop/1176990548

**Mathematical Reviews number (MathSciNet)**

MR1085340

**Zentralblatt MATH identifier**

0723.60043

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62C20: Minimax procedures 90D05

**Keywords**

Secretary problem best-choice problem marriage-problem minimax-optimal stop rule minimax-optimal distribution randomized stop rule

#### Citation

Hill, Theodore P.; Krengel, Ulrich. Minimax-Optimal Stop Rules and Distributions in Secretary Problems. Ann. Probab. 19 (1991), no. 1, 342--353. doi:10.1214/aop/1176990548. https://projecteuclid.org/euclid.aop/1176990548