Open Access
February, 1981 Optimal Stopping with Sampling Cost: The Secretary Problem
Thomas J. Lorenzen
Ann. Probab. 9(1): 167-172 (February, 1981). DOI: 10.1214/aop/1176994519


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.


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Thomas J. Lorenzen. "Optimal Stopping with Sampling Cost: The Secretary Problem." Ann. Probab. 9 (1) 167 - 172, February, 1981.


Published: February, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0449.60033
MathSciNet: MR606808
Digital Object Identifier: 10.1214/aop/1176994519

Primary: 60G40

Keywords: loss function sampling cost , Optimal stopping rules , relative ranks

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 1 • February, 1981
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