Volume
23
Issue
3
Abstract
The venerable Secretary Problem asks how a decision maker (DM) should select one of n candidates, who come up randomly, when the only information available is each candidate’s strict rank within the set of previous applicants. Moreover, DM may select only the current candidate. Our illustrative case has n = 9 candidates, for which the Standard Method is to reject outright the first 3 candidates and then choose the first of the 4th through 8th candidates who is better than all of the first three, or the 9th candidate if none of them is. We compare the Standard Method with two other selection methods that change the conditions under which DM decides:
Reserve Method. Same as the Standard Method, except that the best of the first 3 candidates is held in reserve and chosen if none of the 4th through 8th candidates (Version A) or none of the 4th through 9th candidates (Version B) is better.
Score Method. Each candidate receives a score between 0 and 1; scores are known to be uniformly distributed. In each round, DM decides on a numerical threshold and selects the candidate if she exceeds the threshold. If none does, candidate 9 is selected. We assume that DM chooses thresholds so as to maximize the expected score of the selected candidate.
The Standard Method gives DM a probability of 41% of selecting the best candidate, whereas the Reserve Methods substantially raise this probability to 70% or 74%—depending on whether Version A or B is used—while the Score Method raises it to 55%. Thus, the other methods, especially the Reserve Methods, outperform the Standard Method in selecting the best candidate. But the Score Method requires seeing significantly fewer candidates—an average of 4.3, compared with an average of 6.3 for the Standard Method and 6.3 and 6.7 for the two versions of the Reserve Method.
We also discuss a third criterion, the average rank of the candidate selected, which is about 1.5 for the Reserve Methods and the Score Method but 2.9 for the Standard Method. In sum, the superiority of the alternative methods over the Standard Method reflects the severity of the restrictions placed on the original Secretary Problem, suggesting it is time to revisit the assumptions of this method and consider realistic alternatives.
First Page
235
Last Page
248
Recommended Citation
Brams, Steven J. and Kilgour, D. Marc
(2026)
"Revisiting the Secretary Problem,"
The Mathematics Enthusiast: Vol. 23
:
No.
3
, Article 5.
DOI: https://doi.org/10.54870/1551-3440.1690
Available at:
https://scholarworks.umt.edu/tme/vol23/iss3/5
Digital Object Identifier (DOI)
10.54870/1551-3440.1690
Publisher
University of Montana, Maureen and Mike Mansfield Library
