The Secretary problem is a classical sequential decision-making question that can be succinctly described as follows: a set of rank-ordered applicants are interviewed sequentially for a single position. Once an applicant is interviewed, an immediate and irrevocable decision is made if the person is to be offered the job or not and only applicants observed so far can be used in the decision process. The problem of interest is to identify the stopping rule that maximizes the probability of hiring the highest-ranked applicant. A multiple-choice version of the Secretary problem, known as the Dowry problem, assumes that one is given a fixed integer budget for the total number of selections allowed to choose the best applicant. It has been solved using tools from dynamic programming and optimal stopping theory. We provide the first combinatorial proof for a related new \emph{query-based model} for which we are allowed to solicit the response of an expert to determine if an applicant is optimal. Since the selection criteria differ from those of the Dowry problem we obtain nonidentical expected stopping times. Our result indicates that an optimal strategy is the $(a_s, a_{s-1}, \ldots, a_1)$-strategy, i.e., for the $i^{th}$ selection, where $1 \le i \le s$ and $1 \le j = s+1-i \le s$, we reject the first $a_j$ applicants, wait until the decision of the $(i-1)^{th}$ selection (if $i \ge 2$), and then accept the next applicant whose qualification is better than all previously appeared applicants. Furthermore, our optimal strategy is right-hand based, i.e., the optimal strategies for two models with $s_1$ and $s_2$ selections in total ($s_1 < s_2$) share the same sequence $a_1, a_2, \ldots, a_{s_1}$ when it is viewed from the right. When the total number of applicants tends to infinity, our result agrees with the thresholds obtained by Gilbert and Mosteller.

翻译：秘书问题是一个经典的序贯决策问题，可简述如下：一组按排名排序的应聘者依次接受面试，以填补单一职位空缺。每次面试后，需立即做出不可撤销的决定（是否录用该应聘者），且决策过程仅能依据已面试者的信息。该问题的核心在于确定最优停止规则，以最大化录用到排名第一的应聘者的概率。秘书问题的多选版本称为“嫁妆问题”，假设给定一个固定的整数预算，用于从全体应聘者中选出最佳人选。该问题已通过动态规划和最优停止理论的方法求解。我们首次为一种相关的全新“基于查询模型”提供了组合证明：在该模型中，允许我们征询专家的意见以判断某应聘者是否为最优。由于选择标准不同于嫁妆问题，我们得到了非一致的期望停止时间。结果表明，最优策略是$(a_s, a_{s+1}, \ldots, a_1)$-策略，即对于第$i$次选择（其中$1 \le i \le s$，且$1 \le j = s+1-i \le s$），我们拒绝前$a_j$名应聘者，等待至第$(i-1)$次选择的决策（若$i \ge 2$），然后接受后续出现的、其资质优于此前所有应聘者的第一位候选人。此外，我们的最优策略具有右端基准性，即当从右侧观察时，总选择次数分别为$s_1$和$s_2$（$s_1 < s_2$）的两个模型的最优策略共享相同的序列$a_1, a_2, \ldots, a_{s_1}$。当应聘者总数趋于无穷时，我们的结果与Gilbert和Mosteller所获阈值一致。