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.

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 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.

scholarly journals On the number and sum of near-record observations

2005 ◽  
Vol 37 (03) ◽  
pp. 765-780 ◽  
Author(s):  
N. Balakrishnan ◽  
A.G. Pakes ◽  
A. Stepanov
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Time ◽  
Record Value ◽  
Independent And Identically Distributed

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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

1984 ◽  
Vol 21 (3) ◽  
pp. 521-536 ◽  
Author(s):  
Masami Yasuda
Keyword(s):  
Integral Equation ◽  
Markov Decision Processes ◽  
Random Number ◽  
Scaling Limit ◽  
Decision Processes ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Optimality Equation ◽  
Best Choice Problem ◽  
Markov Decision

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

1984 ◽  
Vol 21 (03) ◽  
pp. 521-536 ◽  
Author(s):  
Masami Yasuda
Keyword(s):  
Integral Equation ◽  
Markov Decision Processes ◽  
Random Number ◽  
Scaling Limit ◽  
Decision Processes ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Optimality Equation ◽  
Best Choice Problem ◽  
Markov Decision

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.

On a best-choice problem by dependent criteria

1994 ◽  
Vol 31 (01) ◽  
pp. 221-234 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Normal Distribution ◽  
Random Sample ◽  
Stopping Rules ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Best Choice Problem

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent.An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.

scholarly journals On the number and sum of near-record observations

2005 ◽  
Vol 37 (3) ◽  
pp. 765-780 ◽  
Author(s):  
N. Balakrishnan ◽  
A.G. Pakes ◽  
A. Stepanov
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Time ◽  
Record Value ◽  
Independent And Identically Distributed

Let X1,X2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables Xi as near-nth-record observations if Xi∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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.

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