The “thirty-seven percent rule” and the secretary problem with relative ranks. (English) Zbl 1401.62128

The theme of this study is to analyse a specific issue, optimal stopping defined in relation to the secretary problem based on the relative ranks (see [T. S. Ferguson, Stat. Sci. 4, No. 3, 282–296 (1989; Zbl 0955.01509)] for a review of various formulations and extensions). The aim of the statistician is to minimize the rank of a selected item (see, e.g. [Y. S. Chow et al., Isr. J. Math. 2, 81–90 (1964; Zbl 0149.14402)] for the solution in the class of stopping rules based on the relative ranks). An explicit (optimal stopping time related to observed relative ranks) and a priori stopping rule that generalizes J. N. Bearden’s [J. Math. Psychol. 50, No. 1, 58–59 (2006; Zbl 1125.90028)] result is investigated (another discussion of Bearden’s suggestion can be found in [A. M. Krieger and E. Samuel-Cahn, Adv. Appl. Probab. 41, No. 4, 1041–1058 (2009; Zbl 1186.62101); K. Szajowski, Sci. Math. Jpn. 69, No. 2, 285–293 (2009; Zbl 1160.62075)]). The problem of minimizing the rank of the applicant hired, when the statistician insists on hiring one of the best \(d\) applicants (\(1\leq d\leq n\)) is analysed. The case \(d = 1\) then yields the classical secretary problem, and the case \(d = n\) corresponds to the expected rank minimization without constrains.


62L15 Optimal stopping in statistics
60C05 Combinatorial probability
60G40 Stopping times; optimal stopping problems; gambling theory
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