Score-Based Secretary Problem

Abstract

In the celebrated “Secretary Problem,” involving n candidates who have applied for a single vacant secretarial position, the employer interviews them one by one in random order and learns their relative ranks. As soon as each interview is over, the employer must either hire the candidate (and stop the process) or reject her (never to be recalled). We consider a variation of this problem where the employer also learns the scores of the already interviewed candidates, which are assumed to be independent and drawn from a known continuous probability distribution. Endowed with this additional information, what strategy should the employer follow in order to maximize his chance of hiring the candidate with the highest score among all n candidates? What is the maximum probability of hiring the best candidate?