Best-choice problems involving recall and uncertainty of selection when the number of observations is random

1984 ◽  
Vol 16 (01) ◽  
pp. 111-130
Author(s):  
Joseph D. Petruccelli
Keyword(s):  
Optimal Stopping ◽  
Random Variable ◽  
Point Of View ◽  
Stopping Rules ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Previous Formulation ◽  
Optimal Stopping Rules ◽  
Bounded Random Variable ◽  
Selection Of

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.

Best-choice problems involving recall and uncertainty of selection when the number of observations is random

1984 ◽  
Vol 16 (1) ◽  
pp. 111-130 ◽  
Author(s):  
Joseph D. Petruccelli
Keyword(s):  
Optimal Stopping ◽  
Random Variable ◽  
Point Of View ◽  
Stopping Rules ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Previous Formulation ◽  
Optimal Stopping Rules ◽  
Bounded Random Variable ◽  
Selection Of

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.

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

1983 ◽  
Vol 20 (1) ◽  
pp. 165-171 ◽  
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.

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

1983 ◽  
Vol 20 (01) ◽  
pp. 165-171 ◽  
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.

Urn sampling distributions giving alternate correspondences between two optimal stopping problems

2016 ◽  
Vol 48 (3) ◽  
pp. 726-743 ◽  
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.

Optimal stopping rules for correlated random walks with a discount

2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Optimal Stopping ◽  
Constant Factor ◽  
Stopping Rules ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
Optimal Rule ◽  
Correlated Random Walks ◽  
Optimal Stopping Rules

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

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