The postdoc variant of the secretary problem

Abstract

ABSTRACT. The classical secretary problem involves sequentially interviewing a pool of n appli-cants with the aim of hiring exactly the best one in the pool—nothing less is good enough. The optimal decision strategy is easy to describe and the probability of success is 1/e. In this paper, we consider a minor variant of this classical problem. We wish to pick not the best but the sec-ond best (the best is going to Harvard). In this case, an explicit solution can be given both for the optimal strategy and the associated optimal success probability. The probability of success is k0(n − k0)/(n(n − 1)) where k0 = bn/2c. Clearly, as n goes to infinity, the probability of success tends to 1/4. Apparently, it is easier to pick the best than the second best. 1. INTRODUCTION. The secretary problem (first popularized by Martin Gardner [4]) is a classical problem in optimal selection. Dynkin [2] and, in slightly different form, Chow et al. [1] were the first to give rigorous treatments. The problem is described as follows. A manager wishes to hire a secretary. An ad is posted and n applicants apply. The candidates are interviewed in random order—nothing is known

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