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.

翻译：考虑一个来自不同大学的候选人招聘过程。具有相同背景的候选人容易排序，但不同背景的候选人比较起来则颇具挑战，后者需要额外的昂贵评估，并可能导致次优的招聘决策。在给定预算的情况下，选择最合格候选人的最优策略是什么？我们通过引入秘书问题的一个新变体来建模上述问题，该变体中依次观测的候选人被分为两个不同的组。对于每个新候选人，决策者观察到其在同一组已观测候选人中的排名，并可以以固定成本获取其在所有已观测候选人中的排名。为解决这一新问题，我们引入并研究了动态双阈值（DDT）算法族。我们证明，在参数选择得当的情况下，随着预算增长，其成功概率迅速收敛至1/e，恢复了经典秘书问题的最优成功概率。最后，针对无记忆算法类，我们提出了非渐近情形下的最优算法，并证明当候选人数量较大时，该算法属于DDT算法族。