The secretary problem with biased arrival order via a Mallows distribution

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 M_n^⁎ items, where [Formula presented], and then to select the first item ranked higher than any of the first M_n^⁎ items (if such an item exists). This yields [Formula presented] as the limiting probability of success. The Mallows distribution with parameter q=1 is the uniform distribution. For the regime [Formula presented], with c>0, the case of weak bias, the optimal strategy occurs with [Formula presented], with the limiting probability of success being [Formula presented]. For the regime [Formula presented], with c>0 and α∈(0,1), the case of moderate bias, the optimal strategy occurs with [Formula presented], with the limiting probability of success being [Formula presented]. For fixed q∈(0,1), the case of strong bias, the optimal strategy occurs with M_n^⁎=n−L where [Formula presented], with limiting probability of success being [Formula presented].