Approximate Results for a Generalized Secretary Problem

Dietz, Chris; Laan, Dinard van der; Ridder, Ad

doi:10.1017/s026996481000032x

Cited by 8 publications

(5 citation statements)

References 9 publications

(24 reference statements)

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“…The PS heuristic outperforms other approximate solutions that have been suggested for these problems. Dietz et al (2011) propose two policies approximating the optimal solution to PMP-2, one of which consists of a single threshold and the other of two thresholds. The former has two free parameters (the relative ranking cutoff to be applied after a position threshold), whereas the latter has four parameters (two rankings and two positions).…”

Section: Results For Non-competitive Optimal Stopping Problemsmentioning

confidence: 99%

Progressive stopping heuristics that excel in individual and competitive sequential search

2022

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We study the performance of heuristics relative to the performance of optimal solutions in the rich domain of sequential search, where the decision to stop the search depends only on the applicant’s relative rank. Considering multiple variants of the secretary problem, that vary from one another in their formulation and method of solution, we find that descriptive heuristics perform well only when the optimal solution prescribes a single threshold value. We show that a computational heuristic originally proposed as an approximate solution to a single variant of the secretary problem performs equally well in many other variants where the optimal solution prescribes multiple threshold values that gradually relax the criterion for stopping the search. Finally, we propose a new heuristic with near optimal performance in a competitive or strategic variant of the secretary problem with multiple employers competing with one another to hire job applicants. Both heuristics share a simple computational component: the ratio of the number of interviewed applicants to the number of those remaining to be searched. We present the subgame-perfect Nash equilibrium for this competitive variant and an algorithm for its computation.

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Section: Results For Non-competitive Optimal Stopping Problemsmentioning

confidence: 99%

Progressive stopping heuristics that excel in individual and competitive sequential search

2022

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“…The known constructions of the optimal sequence via analytic expressions were presented in the similar form as in Theorem 2 but, as we have mentioned above, only in the cases k = 1, 2, 3. For higher values of k, as well as for some other versions of this problem, the algorithms computing the probability of success and the elements of an optimal sequence can be found in various of papers (see [1,2,3,4,5,7,8,11,12,15]). However, in contrast to the analytic solution, these methods apply mechanisms via dynamic or linear programming, and hence only numerically allow to determine the elements of an optimal sequence.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…where the maps r l : [l, n] → R (l ∈ [≤ k]) are defined as in (5). The number of lucky permutations corresponding to w 0 and having no w 0 -thresholds is equal to…”

Section: The Formula For the Probability Of Successmentioning

confidence: 99%

The solution of a generalized secretary problem via analytic expressions

Woryna

2016

J Comb Optim

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Given integers 1 ≤ k < n, the Gusein-Zade version of a generalized secretary problem is to choose one of the k best of n candidates for a secretary, which are interviewing in random order. The stopping rule in the selection is based only on the relative ranks of the successive arrivals. It is known that the best policy can be described by a non-decreasing sequence (s 1 , . . . , s k ) of integers with l ≤ s l < n for every 1 ≤ l ≤ k, and conversely, any such a sequence determines the general structure of the best policy. We found a finite analytic expression for the probability of success when using the optimal policy with a sequence (s 1 , . . . , s k ). We also study the problem of the construction of the optimal sequence, i.e. a sequence which maximizes the corresponding probability of success. We discovered finite analytic expressions which enable to calculate the elements s l of an optimal sequence one by one, from l = k to l = 1. Until now, such expressions were derived separately, and only for the values k ≤ 3.

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“…for a range of different values of k. The recent paper Dietz et al (2011) studies some approximate policies.…”

Section: Problem (A1)mentioning

confidence: 99%

“…Based on general asymptotic results of Mucci (1973), Frank & Samuels (1980) computed numerically lim n→∞ V * n q (k) gz for a range of different values of k. The recent paper Dietz et al (2011) studies some approximate policies.…”

Section: Introductionmentioning

confidence: 99%

A unified approach for solving sequential selection problems

Goldenshluger¹,

Malinovsky²,

Zeevi³

2020

Probab. Surveys

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In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.

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Approximate Results for a Generalized Secretary Problem

Cited by 8 publications

References 9 publications

Progressive stopping heuristics that excel in individual and competitive sequential search

Progressive stopping heuristics that excel in individual and competitive sequential search

The solution of a generalized secretary problem via analytic expressions

A unified approach for solving sequential selection problems

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