We study the secretary problem in which rank-ordered lists are generated by the Mallows model and the goal is to identify the highest-ranked candidate through a sequential interview process which does not allow rejected candidates to be revisited. The main difference between our formulation and existing models is that, during the selection process, we are given a fixed number of opportunities to query an infallible expert whether the current candidate is the highest-ranked or not. If the response is positive, the selection process terminates, otherwise, the search continues until a new potentially optimal candidate is identified. Our optimal interview strategy, as well as the expected number of candidates interviewed and the expected number of queries used, can be determined through the evaluation of well-defined recurrence relations. Specifically, if we are allowed to query $s-1$ times and to make a final selection without querying (thus, making $s$ selections in total) then the optimum scheme is characterized by $s$ thresholds that depend on the parameter $\theta$ of the Mallows distribution but are independent on the maximum number of queries.

翻译：我们研究秘书问题，其中排序列表由Mallows模型生成，目标是通过不允许拒绝候选者重新访问的连续面试过程识别最高排名候选者。我们的模型与现有模型的主要区别在于，在选择过程中，我们有一定数量的固定机会向绝对可靠的专家查询当前候选者是否为最高排名者。若答案为是，则选择过程终止；否则，搜索继续直到识别出新的潜在最优候选者。我们的最优面试策略，以及预期面试候选者数量和预期查询次数，可通过定义明确的递推关系评估来确定。具体而言，如果我们被允许查询$s-1$次并在不查询的情况下做出最终选择（因此总共进行$s$次选择），则最优方案由$s$个阈值刻画，这些阈值依赖于Mallows分布的参数$\theta$，但与最大查询次数无关。