A full-information best-choice problem with finite memory

1986 ◽  
Vol 23 (03) ◽  
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 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.

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.

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.

A competitive best-choice problem with Poisson arrivals

1990 ◽  
Vol 27 (2) ◽  
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.

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

2004 ◽  
Vol 36 (2) ◽  
pp. 398-416 ◽  
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.

The full-information best-choice problem with uniform or gamma horizons

Optimization ◽  
2015 ◽  
Vol 65 (4) ◽  
pp. 765-778 ◽  
Author(s):  
Michael Bendersky ◽  
Israel David
Keyword(s):  
Full Information ◽  
Choice Problem ◽  
Best Choice Problem

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.

scholarly journals Winning Rate in the Full-Information Best-Choice Problem

2007 ◽  
Vol 44 (2) ◽  
pp. 560-565 ◽  
Author(s):  
Alexander V. Gnedin ◽  
Denis I. Miretskiy
Keyword(s):  
Explicit Formula ◽  
Full Information ◽  
Choice Problem ◽  
Best Choice Problem

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem.

Choosing the best of the current crop

1981 ◽  
Vol 13 (3) ◽  
pp. 510-532 ◽  
Author(s):  
Gregory Campbell ◽  
Stephen M. Samuels
Keyword(s):  
Success Probability ◽  
Continuous Distribution ◽  
Training Sample ◽  
Information Value ◽  
Stopping Rules ◽  
Full Information ◽  
Best Choice Problem ◽  
Training Samples ◽  
Intermediate Problem ◽  
Information Problem

A best choice problem is presented which is intermediate between the constraints of the ‘no-information’ problem (observe only the sequence of relative ranks) and the demands of the ‘full-information’ problem (observations from a known continuous distribution). In the intermediate problem prior information is available in the form of a ‘training sample’ of size m and observations are the successive ranks of the n current items relative to their predecessors in both the current and training samples.Optimal stopping rules for this problem depend on m and n essentially only through m + n; and, as m/(m + n) → t, their success probabilities, P*(m, n), converge rapidly to explicitly derived limits p*(t) which are the optimal success probabilities in an infinite version of the problem. For fixed n, P*(m, n) increases with m from the ‘no-information’ optimal success probability to the ‘full-information’ value for sample size n. And as t increases from 0 to 1, p*(t) increases from the ‘no-information’ limit e–1 ≍ 0·37 to the ‘full-information’ limit ≍0·58. In particular p*(0·5) ≍ 0·50.

Sign in / Sign up

Export Citation Format

Share Document