Shelf life of candidates in the generalized secretary problem

The study presents a version of the secretary problem called the duration problem in which the objective is to maximize the time of possession of the relatively best or the second best objects. It is shown that in this duration problem there are threshold numbers such that the optimal strategy is determined by them. When the number of objects tends to infinity the thresholds values are ⌊0.120381N⌋ and ⌊0.417188N⌋, respectively, and the asymptotic mean time of shelf life is 0.403827N.