scholarly journals Secretary Problem with Possible Errors in Observation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1639
Author(s):  
Marek Skarupski
Keyword(s):  
Decision Maker ◽  
Empirical Result ◽  
Optimal Strategy ◽  
Secretary Problem ◽  
Theoretical Studies ◽  
Choice Problem ◽  
Second Best ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Special Cases

The classical secretary problem models a situation in which the decision maker can select or reject in the sequential observation objects numbered by the relative ranks. In theoretical studies, it is known that the strategy is to reject the first 37% of objects and select the next relative best one. However, an empirical result for the problem is that people do not apply the optimal rule. In this article, we propose modeling doubts of decision maker by considering a modification of the secretary problem. We assume that the decision maker can not observe the relative ranks in a proper way. We calculate the optimal strategy in such a problem and the value of the problem. In special cases, we also combine this problem with the no-information best choice problem and a no-information second-best choice problem.

scholarly journals Bayes' Model of the Best-Choice Problem with Disorder

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Vladimir Mazalov ◽  
Evgeny Ivashko
Keyword(s):  
Decision Maker ◽  
Random Variables ◽  
Random Variable ◽  
Random Time ◽  
Choice Problem ◽  
Distribution Law ◽  
Optimal Rule ◽  
Bayes Model ◽  
Best Choice Problem

We consider the best-choice problem with disorder and imperfect observation. The decision-maker observes sequentially a known number of i.i.d random variables from a known distribution with the object of choosing the largest. At the random time the distribution law of observations is changed. The random variables cannot be perfectly observed. Each time a random variable is sampled the decision-maker is informed only whether it is greater than or less than some level specified by him. The decision-maker can choose at most one of the observation. The optimal rule is derived in the class of Bayes' strategies.

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.

The optimal value of markov stopping problems with one-step look ahead policy

1988 ◽  
Vol 25 (3) ◽  
pp. 544-552 ◽  
Author(s):  
Masami Yasuda
Keyword(s):  
Random Number ◽  
Secretary Problem ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Look Ahead ◽  
Optimal Value ◽  
One Step ◽  
The Difference ◽  
The One ◽  
A Charge

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.

A Full-Information Best-Choice Problem with Allowance

1996 ◽  
Vol 10 (1) ◽  
pp. 41-56 ◽  
Author(s):  
Mitsushi Tamaki ◽  
J. George Shanthikumar
Keyword(s):  
Optimal Strategy ◽  
Success Probability ◽  
General Nature ◽  
Regularity Conditions ◽  
Full Information ◽  
Choice Problem ◽  
Underlying Distribution ◽  
Best Choice Problem

This paper considers a variation of the classical full-information best-choice problem. The problem allows success to be obtained even when the best item is not selected, provided the item that is selected is within the allowance of the best item. Under certain regularity conditions on the allowance function, the general nature of the optimal strategy is given as well as an algorithm to determine it exactly. It is also examined how the success probability depends on the allowance function and the underlying distribution of the observed values of the items.

The optimal value of markov stopping problems with one-step look ahead policy

1988 ◽  
Vol 25 (03) ◽  
pp. 544-552 ◽  
Author(s):  
Masami Yasuda
Keyword(s):  
Random Number ◽  
Secretary Problem ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Look Ahead ◽  
Optimal Value ◽  
One Step ◽  
The Difference ◽  
The One ◽  
A Charge

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.

scholarly journals Optimal Stopping with Rank-Dependent Loss

2007 ◽  
Vol 44 (4) ◽  
pp. 996-1011 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Nondecreasing Function ◽  
Stopping Rule ◽  
Choice Problem ◽  
History Dependence ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Complete History ◽  
Independent And Identically Distributed

For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(Rτ)], where Rj is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends 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.

scholarly journals Optimal Stopping with Rank-Dependent Loss

2007 ◽  
Vol 44 (04) ◽  
pp. 996-1011 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Nondecreasing Function ◽  
Stopping Rule ◽  
Choice Problem ◽  
History Dependence ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Complete History ◽  
Independent And Identically Distributed

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.

scholarly journals Variable and sub-optimal responses to a choice problem are a persistent default mode

2019 ◽  
Author(s):  
Amelia R. Hunt ◽  
Warren James ◽  
Josephine Reuther ◽  
Melissa Spilioti ◽  
Eleanor Mackay ◽  
...  
Keyword(s):  
Optimal Strategy ◽  
Decision Rules ◽  
Optimal Solution ◽  
The Other ◽  
Rational Decision ◽  
Choice Problem ◽  
Default Mode ◽  
Potential Benefits ◽  
Simple Logic ◽  
Simple Decision Rule

Here we report persistent choice variability in the presence of a simple decision rule. Two analogous choice problems are presented, both of which involve making decisions about how to prioritize goals. In one version, participants choose a place to stand to throw a beanbag into one of two hoops. In the other, they must choose a place to fixate to detect a target that could appear in one of two boxes. In both cases, participants do not know which of the locations will be the target when they make their choice. The optimal solution to both problems follows the same, simple logic: when targets are close together, standing at/fixating the midpoint is the best choice. When the targets are far apart, accuracy from the midpoint falls, and standing/fixating close to one potential target achieves better accuracy. People do not follow, or even approach, this optimal strategy, despite substantial potential benefits for performance. Two interventions were introduced to try and shift participants from sub-optimal, variable responses to following a fixed, rational rule. First, we put participants into circumstances in which the solution was obvious. After participants correctly solved the problem there, we immediately presented the slightly-less-obvious context. Second, we guided participants to make choices that followed an optimal strategy, and then removed the guidance and let them freely choose. Following both of these interventions, participants immediately returned to a variable, sub-optimal pattern of responding. The results show that while constructing and implementing rational decision rules is possible, making variable responses to choice problems is a strong and persistent default mode. Borrowing concepts from classic animal learning studies, we suggest this default may persist because choice variability can provide opportunities for reinforcement learning.

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