The secretary problem: minimizing the expected rank with I.I.D. random variables

Abstract

n candidates, represented by n i.i.d. continuous random variables X 1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X 1, …, Xn ) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn , and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak , let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.