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An Explicit Formula for the Optimal Gain in the Full-Information Problem of Owning a Relatively Best Object

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Vladimir V. Mazalov  and
Mitsushi Tamaki
Vladimir V. Mazalov*
Affiliation:
Karelian Research Center of the Russian Academy of Science
Mitsushi Tamaki*
Affiliation:
Aichi University
*
∗ Postal address: Institute of Applied Mathematical Research, Karelian Research Center of the Russian Academy of Science, 11 Pushkinskaya Street, Petrozavodsk, 185610, Russia. Email address: vmazalov@trc.karelia.ru
∗∗ Postal address: Department of Business Administration, Aichi University, Nagoya Campus, 370 Kurozasa, Miyoshi, Nishikamo, Aichi 470-0296, Japan. Email address: tamaki@vega.aichi-u.ac.jp
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Abstract

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A full-information version of the secretary problem in which the objective is to maximize the time of possession of a relatively best object is considered both in the case in which only the most recently observed object may be chosen and in the case in which any of the previously observed objects may be chosen. The main purpose of this paper is to obtain, under an optimal rule, both the asymptotic proportional durations when the number of objects tends to infinity, and the expected durations when the number of objects remains finite. The integral expressions for these asymptotic values are derived in both cases: the approximate numerical values are 0.435 in the former (no-recall) case and 0.449 in the latter (recall) case, indicating a surprisingly small difference. The expected values of the optimal stopping times are also obtained from the planar Poisson process analysis.

Keywords

Optimal stoppingsecretary probleme-1 -lawduration problemplanar Poisson processmonotone subsequence problem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 87 - 101
Copyright
© Applied Probability Trust 2006 

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