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 → ∞.

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 → ∞.

Functionals of random mappings: exact and asymptotic results

1991 ◽  
Vol 23 (3) ◽  
pp. 437-455 ◽  
Author(s):  
P. J. Donnelly ◽  
W. J. Ewens ◽  
S. Padmadisastra
Keyword(s):  
Frequency Spectrum ◽  
Exact Results ◽  
Asymptotic Results ◽  
Number Of Components ◽  
Random Mapping ◽  
Random Mappings ◽  
Simulation Results ◽  
Central Tool

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

scholarly journals Opportunity Costs in the Game of Best Choice

10.37236/8322 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Madeline Crews ◽  
Brant Jones ◽  
Kaitlyn Myers ◽  
Laura Taalman ◽  
Michael Urbanski ◽  
...  
Keyword(s):  
Decision Making ◽  
Classical Case ◽  
Secretary Problem ◽  
Sequential Decision Making ◽  
Opportunity Costs ◽  
Sequential Decision ◽  
Probability Of Success ◽  
Uniform Model ◽  
Weighted Model ◽  
Over Time

The game of best choice, also known as the secretary problem, is a model for sequential decision making with many variations in the literature. Notably, the classical setup assumes that the sequence of candidate rankings is uniformly distributed over time and that there is no expense associated with the candidate interviews. Here, we weight each ranking permutation according to the position of the best candidate in order to model costs incurred from conducting interviews with candidates that are ultimately not hired. We compare our weighted model with the classical (uniform) model via a limiting process. It turns out that imposing even infinitesimal costs on the interviews results in a probability of success that is about 28%, as opposed to 1/e (about 37%) in the classical case.

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.

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.

Optimal selection based on relative ranks with a random number of individuals

1979 ◽  
Vol 11 (4) ◽  
pp. 720-736 ◽  
Author(s):  
Jacqueline Gianini-Pettitt
Keyword(s):  
Selection Procedure ◽  
Random Order ◽  
Random Variable ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Asymptotic Results ◽  
Fixed Integer ◽  
Bounded Random Variable ◽  
The Individual ◽  
Number Of Individuals

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

Approximating Markov Chain Approach to Optimal Feedback Control of a Flexible Needle

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Javad Sovizi ◽  
Suren Kumar ◽  
Venkat Krovi
Keyword(s):  
Markov Chain ◽  
Feedback Control ◽  
Stochastic Systems ◽  
Control Policy ◽  
Nonlinear Problems ◽  
Optimal Feedback Control ◽  
Policy Space ◽  
Computationally Efficient ◽  
Insertion Point ◽  
Probability Of Success

Abstract We present a computationally efficient approach for the intra-operative update of the feedback control policy for the steerable needle in the presence of the motion uncertainty. The solution to dynamic programming (DP) equations, to obtain the optimal control policy, is difficult or intractable for nonlinear problems such as steering flexible needle in soft tissue. We use the method of approximating Markov chain to approximate the continuous (and controlled) process with its discrete and locally consistent counterpart. This provides the ground to examine the linear programming (LP) approach to solve the imposed DP problem that significantly reduces the computational demand. A concrete example of the two-dimensional (2D) needle steering is considered to investigate the effectiveness of the LP method for both deterministic and stochastic systems. We compare the performance of the LP-based policy with the results obtained through more computationally demanding algorithm, iterative policy space approximation. Finally, the reliability of the LP-based policy dealing with motion and parametric uncertainties as well as the effect of insertion point/angle on the probability of success is investigated.

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