The secretary problem with biased arrival order via a Mallows distribution

Abstract

We solve the secretary problem in the case that the ranked items arrive in a statistically biased order rather than in uniformly random order. The bias is given by a Mallows distribution with parameter q∈(0,1), so that higher ranked items tend to arrive later and lower ranked items tend to arrive sooner. In the classical problem, the asymptotically optimal strategy is to reject the first Mn⁎ items, where Mn⁎∼ne, and then to select the first item ranked higher than any of the first Mn⁎ items (if such an item exists). This yields 1e as the limiting probability of success. The Mallows distribution with parameter q=1 is the uniform distribution. For the regime qn=1−cn, with c>0, the case of weak bias, the optimal strategy occurs with Mn⁎∼n(1clog⁡(1+ec−1e)), with the limiting probability of success being 1e. For the regime qn=1−cnα, with c>0 and α∈(0,1), the case of moderate bias, the optimal strategy occurs with n−Mn∼nαc, with the limiting probability of success being 1e. For fixed q∈(0,1), the case of strong bias, the optimal strategy occurs with Mn⁎=n−L where L−1L<q≤LL+1, with limiting probability of success being (1−q)qL−1L>1e.