Best-choice Problem Research Articles

A Bayesian Approach to the Best-Choice Problem

Abstract This article considers a version of the so-called best-choice problem. The buyer receives a random sample of price quotations for a good and desires to buy the good at as low a price as possible. After each price quotation is received, the buyer must decide either to accept the price or not, with the objective of maximizing the probability of buying the good at the lowest price. When the distribution of the price quotation is completely known, the optimal buying policy is myopic. The main purpose of this article is to show that this myopic property holds for some interesting price distributions that are updated in a Bayesian manner as successive prices are received.

A full-information best-choice problem with finite memory

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.

A full-information best-choice problem with finite memory

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 p 1(x) or p 2(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.

On a class of best-choice problems

Consideration is given to a new generalized version of the familiar best-choice problem, related to the well-known collective-choice-theory conception of choice function. Certain classes of stopping rules are introduced, and the probability of selecting one of the best (with respect to a given choice function) alternatives is estimated. This work is an approach to choice theory from the viewpoint of statistical sequential analysis.

Asymptotic results for the best-choice problem with a random number of objects

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

Asymptotic results for the best-choice problem with a random number of objects

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

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

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

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

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

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 a Generalization of the Best Choice Problem

Previous article Next article On a Generalization of the Best Choice ProblemM. L. NikolaevM. L. Nikolaevhttps://doi.org/10.1137/1122023PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] M. Gardner, Mathematical games, Sci. Amer., 202 (1960), 150–156, 202 (1960), pp. 172–182 CrossrefGoogle Scholar[2] E. B. Dynkin, The optimal choice of the moment of stopping of a Markov process, Soviet Math. Dokl., 4 (1963), 627–629, (In Russian.) 0242.60018 Google Scholar[3] E. B. Dynkin and , A. A. Yuskevick, Theorems and Problems in Markov Processes, Plenum Press, New York, 1969 CrossrefGoogle Scholar[4] A. N. Širjaev, Statistical sequential analysis, American Mathematical Society, Providence, R.I., 1973iv+174, Vol. 38 of Translations of Math. Monographs MR0350990 Google Scholar[5] S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials, Theory Prob. Applications, 11 (1966), 472–475 10.1137/1111050 LinkGoogle Scholar[6] Gus W. Haggstrom, Optimal sequential procedures when more than one stop is required, Ann. Math. Statist., 38 (1967), 1618–1626 MR0217946 0189.18301 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Improved Online Algorithms for Knapsack and GAP in the Random Order Model17 February 2021 | Algorithmica, Vol. 83, No. 6 Cross Ref Softwarecast: A code-based delivery Manycast scheme in heterogeneous and Opportunistic Ad Hoc NetworksAd Hoc Networks, Vol. 55 Cross Ref The Secretary Problem with a Choice Function27 November 2015 Cross Ref Duration problem with multiple exchangesNumerical Algebra, Control and Optimization, Vol. 2, No. 2 Cross Ref Estimating Optimal Stopping Rules in the Multiple Best Choice Problem with Minimal Summarized Rank via the Cross-Entropy Method Cross Ref Estimating optimal stopping rules in the multiple best choice problem with minimal summarized rank via the Cross-Entropy method Cross Ref Continuation theorems of the extremes under power normalizationJournal of Applied Mathematics and Computing, Vol. 10, No. 1-2 Cross Ref Improved Algorithms and Analysis for Secretary Problems and GeneralizationsMiklos Ajtai, Nimrod Megiddo, and Orli Waarts1 August 2006 | SIAM Journal on Discrete Mathematics, Vol. 14, No. 1AbstractPDF (259 KB)Optimal selection of thek best of a sequence withk stopsMathematical Methods of Operations Research, Vol. 46, No. 2 Cross Ref SELECTING SATISFACTORY SECRETARIES28 June 2008 | Australian Journal of Statistics, Vol. 36, No. 2 Cross Ref Optimal selection of the four best of a sequenceZOR Zeitschrift f� Operations Research Methods and Models of Operations Research, Vol. 38, No. 3 Cross Ref Optimal choice and assignment of the best m of n randomly arriving itemsStochastic Processes and their Applications, Vol. 39, No. 2 Cross Ref Some Extensions of Generalized Secretary ProblemJournal of Information and Optimization Sciences, Vol. 10, No. 2 Cross Ref Asymptotic results for the best-choice problem with a random number of objects14 July 2016 | Journal of Applied Probability, Vol. 21, No. 3 Cross Ref The d choice secretary problem29 March 2007 | Communications in Statistics. Part C: Sequential Analysis, Vol. 2, No. 3 Cross Ref Improved algorithms and analysis for secretary problems and generalizations Cross Ref Volume 22, Issue 1| 1977Theory of Probability & Its Applications History Submitted:15 June 1975Published online:17 July 2006 InformationCopyright © 1977 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1122023Article page range:pp. 187-190ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics