Addressing bias in online selection with limited budget of comparisons

Consider a hiring process with candidates coming from different universities. It is easy to order candidates who have the same background, yet it can be challenging to compare them otherwise. The latter case requires additional costly assessments and can result in sub-optimal hiring decisions. Given an assigned budget, what would be an optimal strategy to select the most qualified candidate? We model the above problem by introducing a new variant of the secretary problem in which sequentially observed candidates are split into two distinct groups. For each new candidate, the decision maker observes its rank among already seen candidates from the same group and can access its rank among all observed candidates at some fixed cost. To tackle this new problem, we introduce and study the family of Dynamic Double Threshold (DDT) algorithms. We show that, with well-chosen parameters, their success probability converges rapidly to 1/e as the budget grows, recovering the optimal success probability from the usual secretary problem. Finally, focusing on the class of memory-less algorithms, we propose an optimal algorithm in the non-asymptotic regime and show that it belongs to the DDT family when the number of candidates is large.