Secretary Problems with Biased Evaluations Using Partial Ordinal Information

The k-secretary problem deals with online selection of at most k numerically scored applicants, where selection decisions are immediate upon their arrival and irrevocable, with the goal of maximizing total score. There is, however, wide prevalence of bias in evaluations of applicants from different demographic groups (e.g., gender, age, race), and the assumption of an algorithm observing their true score is unreasonable in practice. In this work, we propose the poset secretary problem, where selection decisions must be made by observing a partial order over the applicants. This partial order is constructed to account for uncertainty and biases in applicant scores. We assume that each applicant has a fixed score, which is not visible to the algorithm and is consistent with the partial order. Using a random partitioning technique from the matroid secretary literature, we provide order-optimal competitive algorithms for the poset secretary problem and provide matching lower bounds. Further, we develop the theory of thresholding in posets to provide a tight, adaptive-thresholding algorithm under regimes where k grows quickly enough, thus matching the adaptiveness to k shown for the classical k-secretary problem. We then study a special case, in which applicants belong tog disjoint demographic groups and the bias is group specific. We provide competitive algorithms for adversarial and stochastic variants of this special case, including a framework, Gap, for parallelizing any vanilla k-secretary algorithm to the group setting. Finally, we perform a case study on real-world data to demonstrate the responsiveness of our algorithms to data and their impact on selection rates.