## The Annals of Probability

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

Thomas J. Lorenzen

#### 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 in Project Euclid: 19 April 2007

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

Digital Object Identifier
doi:10.1214/aop/1176994519

Mathematical Reviews number (MathSciNet)
MR606808

Zentralblatt MATH identifier
0449.60033

JSTOR