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The secretary problem: minimizing the expected rank with I.I.D. random variables

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

David Assaf  and
Ester Samuel-Cahn
David Assaf*
Affiliation:
The Hebrew University of Jerusalem
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address for both authors: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel.
∗ Postal address for both authors: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel.
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Abstract

n candidates, represented by n i.i.d. continuous random variables X 1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X 1, …, Xn ) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn , and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak , let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.

Keywords

SECRETARY PROBLEMFULL INFORMATIONSTOPPING RULETHRESHOLD RULEFAILURE RATEINDIFFERENCE ON BOUNDARYCALCULUS OF VARIATIONS

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory

Information

Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 828 - 852
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This article was awarded the Joseph Levy prize of the Operations Research Society of Israel, 1994.

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