## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 1 (1981), 1-172

### Optimal Stopping with Sampling Cost: The Secretary Problem

#### Abstract

A secretary problem is an optimal stopping problem based on relative ranks. To the usual formulation of the secretary problem we add a cumulative interview cost function $h(\cdot)$, no longer obtaining "cutoff point" rules. For an appealing form of $h(\cdot)$ we examine the limiting results using the infinite secretary problem. It is shown that the other appealing form of $h(\cdot)$ leads to trivial limiting results. A large class of problems is considered and recursive equations leading to the limiting solution are given. In particular we solve the problem of minimizing expected rank with a linear interview cost function. An approximation to the rank problem with fixed cost $c$ per interview is obtained (for all values of $c$) through the solution of a single differential equation.

#### Article information

**Source**

Ann. Probab. Volume 9, Number 1 (1981), 167-172.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176994519

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176994519

**Mathematical Reviews number (MathSciNet)**

MR606808

**Zentralblatt MATH identifier**

0449.60033

**Subjects**

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

**Keywords**

Optimal stopping rules relative ranks loss function sampling cost

#### Citation

Lorenzen, Thomas J. Optimal Stopping with Sampling Cost: The Secretary Problem. The Annals of Probability 9 (1981), no. 1, 167--172. doi:10.1214/aop/1176994519. http://projecteuclid.org/euclid.aop/1176994519.