A Secretary Problem with a Sliding Window for Recalling Applicants

The Sliding Window Secretary Problem allows a window of choices to the Classical Secretary Problem, in which there is the option to choose the previous K choices immediately prior to the current choice. We consider a case of this sequential choice problem in which the interviewer has a finite, known number of choices and can only discern the relative ranks of choices, and in which every permutation of ranks is equally likely. We examine three cases of the problem: (i) the interviewer has one choice to choose the best applicant; (ii) the interviewer has one choice to choose one of the top two applicants; and (iii) the interviewer has two choices to choose the best applicant. The form of the optimal strategy is shown, the probability of winning as a function of the window size is derived, and the limiting behavior is discussed for all three cases. 1 ar X iv :1 50 8. 07 93 1v 1 [ m at h. PR ] 3 1 A ug 2 01 5