Minimal expected ranks for the secretary problems with uncertain selection (Q2759602)

The paper deals with a version of the secretary problem in which the objective is to minimize the expected absolute rank of the selected applicant. This generalization includes the possibility of an applicant refusing an offer. It is assumed that each applicant accepts an offer of selection with a known probability \(p\), \(p\in (0,1)\), and is independent of the rank of the applicant. The availability of the applicant can be only ascertained by making an offer and stopping when it appears. If no applicant is selected, the risk is \(n\) which corresponds to choosing the worst one. It is shown that the asymptotic value of the problem is \(\prod_{j=1}^\infty \left\{1+\frac{2}{j} \left(\frac{1+pj}{2+p(j-1)} \right) \right\}^{1/(1+pj)}\). It is an extension of the result by \textit{Y. S. Chow}, \textit{S. Moriguti}, \textit{H. Robbins} and \textit{S. M. Samuels} [Isr. J. Math. 2, 81-90 (1964; Zbl 0149.14402)]. See also the papers by \textit{F. T. Bruss} and \textit{T. S. Ferguson} [J. Appl. Probab. 30, No. 3, 616-626 (1993; Zbl 0781.60035)] and \textit{D. Assaf} and \textit{E. Samuel-Cahn} [Adv. Appl. Probab. 28, No. 3, 828-852 (1996; Zbl 0857.60039)]. NEWLINENEWLINENEWLINEThe uncertain selection in the memory-length-one rule, considered by \textit{H. Rubin} and \textit{S. M. Samuels} [Ann. Probab. 5, 627-635 (1977; Zbl 0381.60036)], is examined. An upper bound NEWLINE\[NEWLINEU(p)=\frac{(1+p)^{2 (1+p)/p}-1}{p(1+p)} NEWLINE\]NEWLINE for the limiting minimal expected rank has been obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00065].