scholarly journals Moments of Random Sums and Robbins' Problem of Optimal Stopping

2011 ◽  
Vol 48 (4) ◽  
pp. 1197-1199 ◽  
Author(s):  
Alexander Gnedin ◽  
Alexander Iksanov
Keyword(s):  
Power Function ◽  
Optimal Stopping ◽  
Stopping Rule ◽  
General Context ◽  
Random Sums ◽  
Selection Problems

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

scholarly journals Moments of Random Sums and Robbins' Problem of Optimal Stopping

2011 ◽  
Vol 48 (04) ◽  
pp. 1197-1199 ◽  
Author(s):  
Alexander Gnedin ◽  
Alexander Iksanov
Keyword(s):  
Power Function ◽  
Optimal Stopping ◽  
Stopping Rule ◽  
General Context ◽  
Random Sums ◽  
Selection Problems

Robbins' problem of optimal stopping is that of minimising the expectedrankof an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding thevalueof the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

A Multiple Stopping Problem

1994 ◽  
Vol 8 (2) ◽  
pp. 169-177 ◽  
Author(s):  
J. Preater
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Reward Function ◽  
Optimal Stopping Rule ◽  
Multiple Stopping ◽  
Constant Observation ◽  
Independent And Identically Distributed

In the context of team recruitment, we discuss an optimal multiple stopping problem for an infinite independent and identically distributed sequence, with general reward function and constant observation cost. We establish the existence and nature of an optimal stopping rule. For the particular case where team quality is governed by the fitness of the weakest member, we show that the recruiter should be more discriminating with either a better, or a larger, group of appointees in hand.

scholarly journals The odds algorithm based on sequential updating and its performance

2009 ◽  
Vol 41 (01) ◽  
pp. 131-153 ◽  
Author(s):  
F. Thomas Bruss ◽  
Guy Louchard
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Selection Problems ◽  
Real World Applications ◽  
Sequential Updating ◽  
Indicator Functions ◽  
Sequential Information ◽  
And Performance ◽  
Independent Indicator

LetI1,I2,…,Inbe independent indicator functions on some probability spaceWe suppose that these indicators can be observed sequentially. Furthermore, letTbe the set of stopping times on (Ik),k=1,…,n, adapted to the increasing filtrationwhereThe odds algorithm solves the problem of finding a stopping time τ ∈Twhich maximises the probability of stopping on the lastIk=1, if any. To apply the algorithm, we only need the odds for the events {Ik=1}, that is,rk=pk/(1-pk), whereThe goal of this paper is to offer tractable solutions for the case where thepkare unknown and must be sequentially estimated. The motivation is that this case is important for many real-world applications of optimal stopping. We study several approaches to incorporate sequential information. Our main result is a new version of the odds algorithm based on online observation and sequential updating. Questions of speed and performance of the different approaches are studied in detail, and the conclusiveness of the comparisons allows us to propose always using this algorithm to tackle selection problems of this kind.

scholarly journals OPTIMAL SELECTION OF THE k-TH BEST CANDIDATE

2019 ◽  
Vol 33 (3) ◽  
pp. 327-347
Author(s):  
Yi-Shen Lin ◽  
Shoou-Ren Hsiau ◽  
Yi-Ching Yao
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Optimal Selection ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
The Subject ◽  
Selection Of

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ∞): = lim n→∞p(k, n) is decreasing in k.

OPTIMAL STOPPING IN A STOCHASTIC GAME

2008 ◽  
Vol 23 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Bahar Kaynar
Keyword(s):  
Optimal Stopping ◽  
Stochastic Game ◽  
Stopping Rule ◽  
The Other ◽  
Random Numbers ◽  
Optimal Stopping Rule

In this article we consider a stochastic game in which each player draws one or two random numbers between 0 and 1. Players can decide to stop after the first draw or to continue for a second draw. The decision is made without knowing the other players’ numbers or whether the other players continue for a second draw. The object of the game is to have the highest total score without going over 1. In the article, we will characterize the optimal stopping rule for each player.

Recognizing both the maximum and the second maximum of a sequence

1979 ◽  
Vol 16 (4) ◽  
pp. 803-812 ◽  
Author(s):  
M. Tamaki
Keyword(s):  
Decision Maker ◽  
Optimal Stopping ◽  
Stopping Rule ◽  
Maximum Probability ◽  
Second Best ◽  
Optimal Stopping Rule

We consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants. The optimal stopping rule and the maximum probability of choosing both of them are derived.

Stopping rules for proofreading

1989 ◽  
Vol 26 (02) ◽  
pp. 304-313 ◽  
Author(s):  
T. S. Ferguson ◽  
J. P. Hardwick
Keyword(s):  
Sample Size ◽  
Optimal Stopping ◽  
Random Number ◽  
Poisson Model ◽  
Stopping Rule ◽  
Stopping Rules ◽  
Binomial Model ◽  
Fixed Sample Size ◽  
Fixed Sample ◽  
Optimal Stopping Rule

A manuscript with an unknown random numberMof misprints is subjected to a series of proofreadings in an effort to detect and correct the misprints. On thenthproofreading, each remaining misprint is detected independently with probabilitypn– 1. Each proofreading costs an amountCP&gt; 0, and if one stops afternproofreadings, each misprint overlooked costs an amountcn&gt; 0. Two models are treated based on the distribution ofM.In the Poisson model, the optimal stopping rule is seen to be a fixed sample size rule. In the binomial model, the myopic rule is optimal in many important cases. A generalization is made to problems in which individual misprints may have distinct probabilities of detection and distinct overlook costs.

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