The Infinite Secretary Problem as the Limit of the Finite Problem

Abstract

In a recent paper by J. Gianini and S. M. Samuels an "infinite secretary problem" was formulated: an infinite, countable sequence of rankable individuals (rank 1 = best) arrive at times which are independent and uniformly distributed on [0, 1]. As they arrive, only their relative ranks with respect to their predecessors can be observed. Given an increasing cost function $q(\bullet)$, let $\nu$ be the minimum, among all stopping rules, of the mean of the function $q$ of the actual rank of the individual chosen. Let $\nu(n)$ be the corresponding minimum for a finite secretary problem with $n$ individuals. Then $\lim \nu(n) = \nu$.