Abstract
We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence (s1,s2,...,sm) of critical numbers such that, whenever there remain k choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.
Highlights
Introduction and summaryFerguson et al (1992) were the first to consider a sequential and selection problem referred to as the duration problem, a variation of the classical secretary problem as treated by Gilbert and Mosteller (1966) and others (see Ferguson (1989) and Samuels (1991) for a history and review of the secretary problem)
The multiple–choice duration problem is reformulated as the multiple optimal stopping problem, which has been treated by many authors
We show for the m–choice duration problem that, subject to a condition on the distribution of M, there exists a nonincreasing sequence (s1, s2, . . . , sm) of critical positive integers such that, whenever there remain k choices to be made, the optimal strategy immediately selects a candidate if it appears at or after time sk (1 ≤ k ≤ m)
Summary
Ferguson et al (1992) were the first to consider a sequential and selection problem referred to as the duration problem, a variation of the classical secretary problem as treated by Gilbert and Mosteller (1966) and others (see Ferguson (1989) and Samuels (1991) for a history and review of the secretary problem). For the m–choice duration problem, we choose at most m objects sequentially, and receive unit payoff at each time point as long as the last chosen object remains a candidate, that is, a relatively best object. Ferguson et al (1992) considered another type of problem, the full-information duration problem, where the observations are the actual values of the objects assumed to be independent and identically distributed (iid) from a known distribution and decisions are based on the actual values of the objects They showed that the above equivalence between the best-choice problem and the duration problem holds for the full-information problem. Porosinski (1987); Porosinski (2002) consider the full-information best-choice problem with a random number of objects and Mazalov and Tamaki (2006) and Samuels (2004) the limiting maximum proportional payoff for the full-information one-choice duration problem.
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