The Annals of Probability

The Infinite Secretary Problem with Recall

Amy L. Rocha

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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 $m/n \rightarrow \alpha$.

Article information

Source
Ann. Probab., Volume 21, Number 2 (1993), 898-916.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989273

Digital Object Identifier
doi:10.1214/aop/1176989273

Mathematical Reviews number (MathSciNet)
MR1217571

Zentralblatt MATH identifier
0776.60057

JSTOR
links.jstor.org

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

Keywords
Secretary problem optimal stopping secretary problem with recall best selection best choice problem asymptotic analysis

Citation

Rocha, Amy L. The Infinite Secretary Problem with Recall. Ann. Probab. 21 (1993), no. 2, 898--916. doi:10.1214/aop/1176989273. https://projecteuclid.org/euclid.aop/1176989273


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