## Advances in Applied Probability

### Urn sampling distributions giving alternate correspondences between two optimal stopping problems

Mitsushi Tamaki

#### Abstract

The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let (B,p) denote the best-choice problem and (D,p) the duration problem when the total number N of objects is a bounded random variable with prior p=(p1, p2,...,pn) for a known upper bound n. Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that (D,p) is equivalent to (B,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p(m),m≥0}, i.e. an infinite sequence of priors such that (D,p(m-1)) is equivalent to (B,p(m)) for all m≥1, starting with p(0)=(0,...,0,1). To be more precise, the duration problem is distinguished into (D1,p) or (D2,p), referred to as model 1 or model 2, depending on whether the planning horizon is N or n. The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p[m],m≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as n→∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 726-743.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296312

Mathematical Reviews number (MathSciNet)
MR3568889

Zentralblatt MATH identifier
1351.60051

#### Citation

Tamaki, Mitsushi. Urn sampling distributions giving alternate correspondences between two optimal stopping problems. Adv. in Appl. Probab. 48 (2016), no. 3, 726--743. https://projecteuclid.org/euclid.aap/1474296312

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