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.

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.

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.

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.

A full-information best-choice problem with finite memory

1986 ◽  
Vol 23 (3) ◽  
pp. 718-735 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Simple Algorithm ◽  
Full Information ◽  
Double Sequences ◽  
Choice Problem ◽  
Optimal Procedure ◽  
Best Choice Problem ◽  
Finite Memory ◽  
Special Case

n i.i.d. random variables with known continuous distribution are observed sequentially with the objective of selecting the largest. This paper considers the finite-memory case which, at each stage, allows a solicitation of anyone of the last m observations as well as of the present one. If the (k – t)th observation with value x is solicited at the k th stage, the probability of successful solicitation is p1(x) or p2(x) according to whether t = 0 or 1 ≦ t ≦ m. The optimal procedure is shown to be characterized by the double sequences of decision numbers. A simple algorithm for calculating the decision numbers and the probability of selecting the largest is obtained in a special case.

Negative binomial sums of random variables and discounted reward processes

1998 ◽  
Vol 35 (3) ◽  
pp. 589-599
Author(s):  
William L. Cooper
Keyword(s):  
Differential Equation ◽  
Negative Binomial ◽  
Infinite Horizon ◽  
Random Variables ◽  
Random Variable ◽  
Random Time ◽  
Sums Of Random Variables ◽  
Total Reward ◽  
Binomial Random Variable ◽  
Binomial Sums

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

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.

Two Methods for the Expectation in Envelopes Pairing Problem

2015 ◽  
Vol 713-715 ◽  
pp. 2016-2019
Author(s):  
Lei Qin ◽  
Kang Zhou
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mathematical Induction ◽  
General Situation ◽  
Distribution Law ◽  
Pairing Problem

This paper introduces two methods for solving the expectation in Envelopes Pairing Problem, one is that the random variable is expressed as the sum of n random variables and then solves its expectation; another method is put forward by this paper, that is, firstly the distribution law is derived from simple cases, then considers problems of general situation, gets the expectation of conjecture by simple cases, finally proves the conjecture with mathematical induction.

A competitive best-choice problem with Poisson arrivals

1990 ◽  
Vol 27 (02) ◽  
pp. 333-342 ◽  
Author(s):  
E. G. Enns ◽  
E. Z. Ferenstein
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Optimal Strategies ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Decision Scheme ◽  
Best Choice Problem ◽  
Poisson Stream ◽  
Poisson Arrivals ◽  
Independent And Identically Distributed

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0,T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.

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.

On Random Variables which have the same Distribution as their Reciprocals

1965 ◽  
Vol 8 (6) ◽  
pp. 819-824 ◽  
Author(s):  
V. Seshadri
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Interesting Property ◽  
Distribution Law ◽  
Remarkable Property ◽  
F Distribution

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

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