Optimal Stopping with Rank-Dependent Loss

For τ, a stopping rule adapted to a sequence ofnindependent and identically distributed observations, we define the loss to be E[q(Rτ)], whereRjis the rank of thejth observation andqis a nondecreasing function of the rank. This setting covers both the best-choice problem, withq(r) =1(r> 1), and Robbins' problem, withq(r) =r. Asntends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

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Why do these quite different best-choice problems have the same solutions?

Advances in Applied Probability ◽

10.1239/aap/1086957578 ◽

2004 ◽

Vol 36 (2) ◽

pp. 398-416 ◽

Cited By ~ 13

Author(s):

Stephen M. Samuels

Keyword(s):

Poisson Process ◽

Process Model ◽

Partial Information ◽

Stopping Rule ◽

Good Explanation ◽

Full Information ◽

Choice Problem ◽

Best Choice Problem ◽

Current Observation ◽

Information Problem

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

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Why do these quite different best-choice problems have the same solutions?

Advances in Applied Probability ◽

10.1017/s0001867800013537 ◽

2004 ◽

Vol 36 (02) ◽

pp. 398-416 ◽

Cited By ~ 7

Author(s):

Stephen M. Samuels

Keyword(s):

Poisson Process ◽

Process Model ◽

Partial Information ◽

Stopping Rule ◽

Good Explanation ◽

Full Information ◽

Choice Problem ◽

Best Choice Problem ◽

Current Observation ◽

Information Problem

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

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On the full information best-choice problem

Journal of Applied Probability ◽

10.2307/3215349 ◽

1996 ◽

Vol 33 (3) ◽

pp. 678-687 ◽

Cited By ~ 23

Author(s):

Alexander V. Gnedin

Keyword(s):

Poisson Process ◽

Optimal Stopping ◽

Infinite Sequence ◽

Full Information ◽

Optimal Stopping Problem ◽

Choice Problem ◽

Best Choice Problem ◽

Finite Problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

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On the full information best-choice problem

Journal of Applied Probability ◽

10.1017/s0021900200100117 ◽

1996 ◽

Vol 33 (03) ◽

pp. 678-687 ◽

Cited By ~ 9

Author(s):

Alexander V. Gnedin

Keyword(s):

Poisson Process ◽

Optimal Stopping ◽

Infinite Sequence ◽

Full Information ◽

Optimal Stopping Problem ◽

Choice Problem ◽

Best Choice Problem ◽

Finite Problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

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Estimating Optimal Stopping Rules in the Multiple Best Choice Problem with Minimal Summarized Rank via the Cross-Entropy Method

Evolutionary Learning and Optimization - Exploitation of Linkage Learning in Evolutionary Algorithms ◽

10.1007/978-3-642-12834-9_11 ◽

2010 ◽

pp. 227-241 ◽

Cited By ~ 1

Author(s):

T. V. Polushina

Keyword(s):

Optimal Stopping ◽

Entropy Method ◽

Cross Entropy ◽

Stopping Rules ◽

Choice Problem ◽

Best Choice Problem ◽

Cross Entropy Method ◽

Optimal Stopping Rules ◽

The Cross

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Urn sampling distributions giving alternate correspondences between two optimal stopping problems

Advances in Applied Probability ◽

10.1017/apr.2016.25 ◽

2016 ◽

Vol 48 (3) ◽

pp. 726-743 ◽

Cited By ~ 1

Author(s):

Mitsushi Tamaki

Keyword(s):

Optimal Stopping ◽

Random Variable ◽

Planning Horizon ◽

Information Model ◽

Secretary Problem ◽

Choice Problem ◽

Sampling Distributions ◽

Best Choice Problem ◽

Optimal Stopping Problems ◽

Bounded Random Variable

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.

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A Multiple Stopping Problem

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003314 ◽

1994 ◽

Vol 8 (2) ◽

pp. 169-177 ◽

Cited By ~ 2

Author(s):

J. Preater

Keyword(s):

Optimal Stopping ◽

Stopping Rule ◽

Reward Function ◽

Optimal Stopping Rule ◽

Multiple Stopping ◽

Constant Observation ◽

Independent And Identically Distributed

In the context of team recruitment, we discuss an optimal multiple stopping problem for an infinite independent and identically distributed sequence, with general reward function and constant observation cost. We establish the existence and nature of an optimal stopping rule. For the particular case where team quality is governed by the fitness of the weakest member, we show that the recruiter should be more discriminating with either a better, or a larger, group of appointees in hand.

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On a best choice problem related to the Poisson process

Strategies for Sequential Search and Selection in Real Time - Contemporary Mathematics ◽

10.1090/conm/125/1160609 ◽

1992 ◽

pp. 59-64 ◽

Cited By ~ 6

Author(s):

Alexander V. Gnedin ◽

Minoru Sakaguchi

Keyword(s):

Poisson Process ◽

Choice Problem ◽

Best Choice Problem

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On the best-choice problem when the number of observations is random

Journal of Applied Probability ◽

10.2307/3213731 ◽

1983 ◽

Vol 20 (1) ◽

pp. 165-171 ◽

Cited By ~ 10

Author(s):

Joseph D. Petruccelli

Keyword(s):

Optimal Stopping ◽

Random Variable ◽

Stopping Rules ◽

Necessary And Sufficient Condition ◽

Choice Problem ◽

Best Choice Problem ◽

Optimal Stopping Rule ◽

Optimal Stopping Rules ◽

Bounded Random Variable ◽

Necessary And Sufficient

We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.

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