A secretary problem with backward solicitation and uncertain employment

1983 ◽  
Vol 20 (04) ◽  
pp. 891-896
Author(s):  
K. I. Choe ◽  
D. S. Bai
Keyword(s):  
Optimal Strategy ◽  
Secretary Problem ◽  
Fixed Probability

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.

A secretary problem with backward solicitation and uncertain employment

1983 ◽  
Vol 20 (4) ◽  
pp. 891-896 ◽  
Author(s):  
K. I. Choe ◽  
D. S. Bai
Keyword(s):  
Optimal Strategy ◽  
Secretary Problem ◽  
Fixed Probability

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.

Exact results for a secretary problem

1996 ◽  
Vol 33 (03) ◽  
pp. 630-639 ◽  
Author(s):  
M. P. Quine ◽  
J. S. Law
Keyword(s):  
Markov Chain ◽  
Optimal Strategy ◽  
Secretary Problem ◽  
Exact Results ◽  
Asymptotic Results ◽  
Backwards Induction ◽  
Imbedded Markov Chain ◽  
Probability Of Success

We consider the following secretary problem: items ranked from 1 to n are randomly selected without replacement, one at a time, and to ‘win' is to stop at an item whose overall rank is less than or equal to s, given only the relative ranks of the items drawn so far. Our method of analysis is based on the existence of an imbedded Markov chain and uses the technique of backwards induction. In principal the approach can be used to give exact results for any value of s; we do the working for s = 3. We give exact results for the optimal strategy, the probability of success and the distribution of T, and the total number of draws when the optimal strategy is implemented. We also give some asymptotic results for these quantities as n → ∞.

scholarly journals The Noisy Secretary Problem and Some Results on Extreme Concomitant Variables

2012 ◽  
Vol 49 (3) ◽  
pp. 821-837 ◽  
Author(s):  
Abba M. Krieger ◽  
Ester Samuel-Cahn
Keyword(s):  
Optimal Strategy ◽  
Noisy Data ◽  
Optimal Strategies ◽  
Secretary Problem ◽  
Distribution Free ◽  
Concomitant Variables ◽  
Noisy Observation ◽  
Independent And Identically Distributed

The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with a nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly different when the qualities of the secretaries are observed with noise. If there is no noise then the only information that is needed is whether an observation is the best among those already observed. Since the observations are assumed to be independent and identically distributed, the solution to this problem is distribution free. In the case of noisy data, the results are no longer distribution free. Furthermore, we need to know the rank of the noisy observation among those already observed. Finally, the probability of finding the best secretary often goes to 0 as the number of observations, n, goes to ∞. The results heavily depend on the behavior of pn, the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results.

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 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.

scholarly journals The Noisy Secretary Problem and Some Results on Extreme Concomitant Variables

2012 ◽  
Vol 49 (03) ◽  
pp. 821-837
Author(s):  
Abba M. Krieger ◽  
Ester Samuel-Cahn
Keyword(s):  
Optimal Strategy ◽  
Noisy Data ◽  
Optimal Strategies ◽  
Secretary Problem ◽  
Distribution Free ◽  
Concomitant Variables ◽  
Noisy Observation ◽  
Independent And Identically Distributed

The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with a nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly different when the qualities of the secretaries are observed with noise. If there is no noise then the only information that is needed is whether an observation is the best among those already observed. Since the observations are assumed to be independent and identically distributed, the solution to this problem is distribution free. In the case of noisy data, the results are no longer distribution free. Furthermore, we need to know the rank of the noisy observation among those already observed. Finally, the probability of finding the best secretary often goes to 0 as the number of observations, n, goes to ∞. The results heavily depend on the behavior of p n , the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results.

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.

scholarly journals Secretary Problem with Possible Errors in Observation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1639
Author(s):  
Marek Skarupski
Keyword(s):  
Decision Maker ◽  
Empirical Result ◽  
Optimal Strategy ◽  
Secretary Problem ◽  
Theoretical Studies ◽  
Choice Problem ◽  
Second Best ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Special Cases

The classical secretary problem models a situation in which the decision maker can select or reject in the sequential observation objects numbered by the relative ranks. In theoretical studies, it is known that the strategy is to reject the first 37% of objects and select the next relative best one. However, an empirical result for the problem is that people do not apply the optimal rule. In this article, we propose modeling doubts of decision maker by considering a modification of the secretary problem. We assume that the decision maker can not observe the relative ranks in a proper way. We calculate the optimal strategy in such a problem and the value of the problem. In special cases, we also combine this problem with the no-information best choice problem and a no-information second-best choice problem.

Exact results for a secretary problem

1996 ◽  
Vol 33 (3) ◽  
pp. 630-639 ◽  
Author(s):  
M. P. Quine ◽  
J. S. Law
Keyword(s):  
Markov Chain ◽  
Optimal Strategy ◽  
Secretary Problem ◽  
Exact Results ◽  
Asymptotic Results ◽  
Backwards Induction ◽  
Imbedded Markov Chain ◽  
Probability Of Success

We consider the following secretary problem: items ranked from 1 to n are randomly selected without replacement, one at a time, and to ‘win' is to stop at an item whose overall rank is less than or equal to s, given only the relative ranks of the items drawn so far. Our method of analysis is based on the existence of an imbedded Markov chain and uses the technique of backwards induction. In principal the approach can be used to give exact results for any value of s; we do the working for s = 3. We give exact results for the optimal strategy, the probability of success and the distribution of T, and the total number of draws when the optimal strategy is implemented. We also give some asymptotic results for these quantities as n → ∞.

Adaptive approach to some stopping problems

1985 ◽  
Vol 22 (3) ◽  
pp. 644-652 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Strategy ◽  
Value Function ◽  
Secretary Problem ◽  
Adaptive Approach ◽  
Parking Problem ◽  
Monotonicity Properties ◽  
The Value Function

This paper mainly considers the adaptive version of two typical stopping problems, i.e., the parking problem and the secretary problem with refusal. In the first problem, while driving towards a destination, we observe the successive parking places and note whether or not they are occupied. Unoccupied spaces are assumed to occur independently, with probability p. The second problem is to select the best applicant from a population, where each applicant refuses an offer with probability 1 – p. We assume beta prior for p in advance. As time progresses, we update our belief for p in a Bayesian manner based on the observed states of the process. We derive several monotonicity properties of the value function and characterize the optimal strategy in either problem. We also attempt to relax the same probability condition in the classical parking problem.

Adaptive approach to some stopping problems

1985 ◽  
Vol 22 (03) ◽  
pp. 644-652 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Strategy ◽  
Value Function ◽  
Secretary Problem ◽  
Adaptive Approach ◽  
Parking Problem ◽  
Monotonicity Properties ◽  
The Value Function

This paper mainly considers the adaptive version of two typical stopping problems, i.e., the parking problem and the secretary problem with refusal. In the first problem, while driving towards a destination, we observe the successive parking places and note whether or not they are occupied. Unoccupied spaces are assumed to occur independently, with probability p. The second problem is to select the best applicant from a population, where each applicant refuses an offer with probability 1 – p. We assume beta prior for p in advance. As time progresses, we update our belief for p in a Bayesian manner based on the observed states of the process. We derive several monotonicity properties of the value function and characterize the optimal strategy in either problem. We also attempt to relax the same probability condition in the classical parking problem.

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