The secretary problem: optimal selection from two streams of candidates

1992 ◽  
pp. 65-75
Author(s):  
Z. Govindarajulu
Keyword(s):  
Secretary Problem ◽  
Optimal Selection

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.

scholarly journals Odds Theorem with Multiple Selection Chances

2010 ◽  
Vol 47 (04) ◽  
pp. 1093-1104 ◽  
Author(s):  
Katsunori Ano ◽  
Hideo Kakinuma ◽  
Naoto Miyoshi
Keyword(s):  
Finite Number ◽  
Closed Form ◽  
Selection Rule ◽  
Random Variables ◽  
Secretary Problem ◽  
Optimal Selection ◽  
Maximum Probability ◽  
Optimal Rule ◽  
Interesting Implication

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

Optimal selection based on relative rank (the “secretary problem”)

1964 ◽  
Vol 2 (2) ◽  
pp. 81-90 ◽  
Author(s):  
Y. S. Chow ◽  
S. Moriguti ◽  
H. Robbins ◽  
S. M. Samuels
Keyword(s):  
Secretary Problem ◽  
Optimal Selection ◽  
Relative Rank

On an optimal selection problem of Cowan and Zabczyk

1987 ◽  
Vol 24 (4) ◽  
pp. 918-928 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Intensity Function ◽  
Poisson Processes ◽  
Secretary Problem ◽  
Selection Problem ◽  
Recall Condition ◽  
Optimal Selection ◽  
Multiplicative Constant ◽  
Inhomogeneous Poisson Process

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

scholarly journals Odds Theorem with Multiple Selection Chances

2010 ◽  
Vol 47 (4) ◽  
pp. 1093-1104 ◽  
Author(s):  
Katsunori Ano ◽  
Hideo Kakinuma ◽  
Naoto Miyoshi
Keyword(s):  
Finite Number ◽  
Closed Form ◽  
Selection Rule ◽  
Random Variables ◽  
Secretary Problem ◽  
Optimal Selection ◽  
Maximum Probability ◽  
Optimal Rule ◽  
Interesting Implication

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

On an optimal selection problem of Cowan and Zabczyk

1987 ◽  
Vol 24 (04) ◽  
pp. 918-928 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Intensity Function ◽  
Poisson Processes ◽  
Secretary Problem ◽  
Selection Problem ◽  
Recall Condition ◽  
Optimal Selection ◽  
Multiplicative Constant ◽  
Inhomogeneous Poisson Process

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

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