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.

A secretary problem with uncertain employment

1975 ◽  
Vol 12 (03) ◽  
pp. 620-624 ◽  
Author(s):  
M. H. Smith
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Maximum Probability ◽  
Fixed Probability ◽  
Optimal Stopping Rule

A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 &lt; p &lt; 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.

A secretary problem with uncertain employment

1975 ◽  
Vol 12 (3) ◽  
pp. 620-624 ◽  
Author(s):  
M. H. Smith
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Maximum Probability ◽  
Fixed Probability ◽  
Optimal Stopping Rule

A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.

A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

2014 ◽  
Vol 51 (03) ◽  
pp. 885-889 ◽  
Author(s):  
Tomomi Matsui ◽  
Katsunori Ano
Keyword(s):  
Lower Bound ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
Optimal Probability ◽  
Optimal Stopping Theory ◽  
Special Case

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m &lt; N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

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.

Optimal stopping on trajectories and the ballot problem

2001 ◽  
Vol 38 (04) ◽  
pp. 946-959 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
Ballot Problem

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let P n and M n denote the respective numbers of plus balls and minus balls drawn by time n and define Z 0 = 0, Z n = P n - M n , 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.

scholarly journals Sum the Multiplicative Odds to One and Stop

2010 ◽  
Vol 47 (03) ◽  
pp. 761-777 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Process Approach ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Full Information ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Asymptotic Results ◽  
Optimal Stopping Rule

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m &lt; n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

scholarly journals A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

2014 ◽  
Vol 51 (3) ◽  
pp. 885-889 ◽  
Author(s):  
Tomomi Matsui ◽  
Katsunori Ano
Keyword(s):  
Lower Bound ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
Optimal Probability ◽  
Optimal Stopping Theory ◽  
Special Case

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

Recognizing both the maximum and the second maximum of a sequence

1979 ◽  
Vol 16 (04) ◽  
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.

Optimal stopping on trajectories and the ballot problem

2001 ◽  
Vol 38 (4) ◽  
pp. 946-959 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
Ballot Problem

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let Pn and Mn denote the respective numbers of plus balls and minus balls drawn by time n and define Z0 = 0, Zn = Pn - Mn, 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.

scholarly journals Sum the Multiplicative Odds to One and Stop

2010 ◽  
Vol 47 (3) ◽  
pp. 761-777 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Process Approach ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Full Information ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Asymptotic Results ◽  
Optimal Stopping Rule

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

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