scholarly journals Optimal Stopping in the Balls-and-Bins Problem

2021 ◽  
Vol 14 ◽  
pp. 183-191
Author(s):  
Anna A. Ivashko ◽  
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Expected Payoff ◽  
Allocation Model ◽  
Job Allocation

This paper considers a multistage balls-and-bins problem with optimal stopping connected with the job allocation model. There are N steps. The player drops balls (tasks) randomly one at a time into available bins (servers). The game begins with only one empty bin. At each step, a new bin can appear with probability p. At step n (n = 1, . . . ,N), the player can choose to stop and receive the payoff or continue the process and move to the next step. If the player stops, then he/she gets 1 for every bin with exactly one ball and loses 1/2 for every bin with two or more balls. Empty bins do not count. At the last step, the player must stop the process. The player's aim is to find the stopping rule which maximizes the expected payoff. The optimal payoff at each step are calculated. An approximate strategy depending on the number of steps is proposed. It is demonstrated that the payo when using this strategy is close to the optimal payoff.

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 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.

scholarly journals On the optimal stopping problems with monotone thresholds

2015 ◽  
Vol 52 (4) ◽  
pp. 926-940 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Expected Value ◽  
Choice Problem ◽  
Expected Payoff ◽  
Remarkable Feature ◽  
Best Choice Problem ◽  
Optimal Probability ◽  
Optimal Stopping Problems

As a class of optimal stopping problems with monotone thresholds, we define the candidate-choice problem (CCP) and derive two formulae for calculating its expected payoff. We apply the first formula to a particular CCP, i.e. the best-choice duration problem treated by Ferguson et al. (1992). The recall case is also examined as a comparison. We also derive the distribution of the stopping time of the CCP and find, as a by-product, that the best-choice problem has a remarkable feature in that the optimal probability of choosing the best is just the expected value of the (proportional) stopping time. The similarity between the best-choice duration problem and the best-choice problem with uniform freeze studied by Samuel-Cahn (1996) is recognized.

scholarly journals On the optimal stopping problems with monotone thresholds

2015 ◽  
Vol 52 (04) ◽  
pp. 926-940 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Expected Value ◽  
Choice Problem ◽  
Expected Payoff ◽  
Remarkable Feature ◽  
Best Choice Problem ◽  
Optimal Probability ◽  
Optimal Stopping Problems

As a class of optimal stopping problems with monotone thresholds, we define the candidate-choice problem (CCP) and derive two formulae for calculating its expected payoff. We apply the first formula to a particular CCP, i.e. the best-choice duration problem treated by Ferguson et al. (1992). The recall case is also examined as a comparison. We also derive the distribution of the stopping time of the CCP and find, as a by-product, that the best-choice problem has a remarkable feature in that the optimal probability of choosing the best is just the expected value of the (proportional) stopping time. The similarity between the best-choice duration problem and the best-choice problem with uniform freeze studied by Samuel-Cahn (1996) is recognized.

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.

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.

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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