Generalizing the Secretary Problem with Rank-dependent Rejection Probability

Abstract

Tamaki and Ohno [9] consider the secretary problem with a rank-dependent rejection probability which can be stated as follows: n applicants appear one at a time in random order with all n! permutations equally likely. If we could observe them all, we could rank them absolutely with no ties from best(rank 1) to worst(rank n). However, each time an applicant appears, we observe the rank of the applicant relative to those preceding her and decide, based on the observed rank(s), whether to make an offer of employment to the current applicant or to pass over her and observe the next. The j-th ranked applicant is assumed to reject(accept) an offer with a fixed probability q _j (p _j = 1 − q _j ), 1 ≤ j ≤ n, independent of the other applicants. No recall of the previous applicants is allowed and the process terminates once an applicant is selected, i.e., an offer is acc epted or the final time is reached with no applicant selected. The objective of the problem is to find a policy that will maximize the probability of selecting the overall best.