Approximate Results for a Generalized Secretary Problem

A version of the classical secretary problem is studied, in which one is interested in selecting one of the <I>b</I> best out of a group of <I>n</I> differently ranked persons who are presented one by one in a random order. It is assumed that <I>b</I> is bigger than or equal to 1 is a preassigned number. It is known, already for a long time, that for the optimal policy one needs to compute <I>b</I> position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as <I>n</I> goes to infinity) results, which show that the double-level policy is an extremely accurate approximation.