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April, 1993 The Infinite Secretary Problem with Recall
Amy L. Rocha
Ann. Probab. 21(2): 898-916 (April, 1993). DOI: 10.1214/aop/1176989273

Abstract

The infinite secretary problem, in which an infinite number of rankable items arrive at times which are i.i.d., uniform on (0, 1), is modified to allow for a fixed period of recall of length $\alpha, 0 \leq \alpha \leq 1$. The goal is to find the maximum probability of best choice, $v = v(\alpha)$, as well as an optimal stopping time $\tau^\ast = \tau^\ast(\alpha)$. A differential-delay equation is derived, the solution of which yields $v(\alpha)$ and $\tau^\ast(\alpha)$, the latter given in terms of a constant $t^\ast \lbrack = t^\ast(\alpha)\rbrack$. For $\alpha \geq 1/2$, the complete solution to the problem is obtained. For $0 < \alpha < 1/2, v(\alpha)$ cannot be put in closed form, so upper and lower bounds for $v(\alpha)$ and $t^\ast(\alpha)$ are obtained and are investigated for $\alpha$ near 0 and near 1/2, where the solutions are known. We also find asymptotic expansions of $v(\alpha)$ and $t^\ast(\alpha)$ about $\alpha = 0$ and $\alpha = 1/2$. Finally, the solution to the finite, $n$-item length-$m$ recall problem introduced by Smith and Deely is shown to converge to the solution of the infinite problem when $m/n \rightarrow \alpha$.

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Amy L. Rocha. "The Infinite Secretary Problem with Recall." Ann. Probab. 21 (2) 898 - 916, April, 1993. https://doi.org/10.1214/aop/1176989273

Information

Published: April, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0776.60057
MathSciNet: MR1217571
Digital Object Identifier: 10.1214/aop/1176989273

Subjects:
Primary: 60G40
Secondary: 62L15

Keywords: asymptotic analysis , best choice problem , best selection , Optimal stopping , secretary problem , secretary problem with recall

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 2 • April, 1993
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