Shelf Life of Candidates in the Generalized Secretary Problem

A version of the secretary problem called the duration problem, in which the objective is to maximize the time of possession of relatively best objects or the second best, is treated. It is shown that in this duration problem there are threshold numbers $(k_1^{\star },k_2^{\star })$ such that the optimal strategy immediately selects a relatively best object if it appears after time $k_1^{\star }$ and a relatively second best object if it appears after moment $k_2^{\star }$. When number of objects tends to infinity the thresholds values are $\lfloor 0.417188N\rfloor$ and $\rfloor 0.120381N\rfloor$, respectively. The asymptotic mean time of shelf life of the object is $0.403827N$.