scholarly journals Compare the ratio of symmetric polynomials of odds to one and stop

2017 ◽  
Vol 54 (1) ◽  
pp. 12-22
Author(s):  
Tomomi Matsui ◽  
Katsunori Ano
Keyword(s):  
Lower Bound ◽  
Optimal Stopping ◽  
Optimization Technique ◽  
Secretary Problem ◽  
Symmetric Polynomials ◽  
Threshold Strategy ◽  
Optimal Stopping Rule ◽  
Special Cases ◽  
Unified View ◽  
Special Case

AbstractIn this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last ℓ successes, given a sequence of independent Bernoulli trials of length N, where k and ℓ are predetermined integers satisfying 1≤k≤ℓ<N. This problem includes some odds problems as special cases, e.g. Bruss’ odds problem, Bruss and Paindaveine’s problem of selecting the last ℓ successes, and Tamaki’s multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton’s inequalities and optimization technique, which gives a unified view to the previous works.

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.

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.

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.

Recognizing the maximum of a random sequence based on relative rank with backward solicitation

1974 ◽  
Vol 11 (03) ◽  
pp. 504-512 ◽  
Author(s):  
Mark C. K. Yang
Keyword(s):  
General Formula ◽  
Optimal Stopping ◽  
Random Sequence ◽  
Geometric Probability ◽  
Secretary Problem ◽  
Stopping Rules ◽  
Relative Rank ◽  
Special Cases ◽  
Optimal Stopping Rules

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

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.

Minimizing the expected rank with full information

1993 ◽  
Vol 30 (03) ◽  
pp. 616-626 ◽  
Author(s):  
F. Thomas Bruss ◽  
Thomas S. Ferguson
Keyword(s):  
Finite Number ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Full Information ◽  
Asymptotically Optimal ◽  
A Value ◽  
Optimal Stopping Rule ◽  
Optimal Value ◽  
History Of ◽  
Threshold Rule

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.

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.

Recognizing the maximum of a random sequence based on relative rank with backward solicitation

1974 ◽  
Vol 11 (3) ◽  
pp. 504-512 ◽  
Author(s):  
Mark C. K. Yang
Keyword(s):  
General Formula ◽  
Optimal Stopping ◽  
Random Sequence ◽  
Geometric Probability ◽  
Secretary Problem ◽  
Stopping Rules ◽  
Relative Rank ◽  
Special Cases ◽  
Optimal Stopping Rules

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

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.

Minimizing the expected rank with full information

1993 ◽  
Vol 30 (3) ◽  
pp. 616-626 ◽  
Author(s):  
F. Thomas Bruss ◽  
Thomas S. Ferguson
Keyword(s):  
Finite Number ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Full Information ◽  
Asymptotically Optimal ◽  
A Value ◽  
Optimal Stopping Rule ◽  
Optimal Value ◽  
History Of ◽  
Threshold Rule

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.

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