We consider two variations of the classical secretary problem. * A variation of the returning secretary problem where each interviewee may appear a second time with a fixed probability p. The decision-maker observes interviewees sequentially and must choose whether to accept or reject each appearance. We characterize the optimal threshold rule and examine its dependence on the reappearance probability p, highlighting how additional information from repeated appearances improves selection performance. * A variation of the secretary problem in which success is defined as selecting any one of the top three interviewees rather than the single best. Interviewees are observed sequentially in random order, and decisions are irreversible. We estimated the success probability under this relaxed success criterion using the threshold strategy of the classical secretary problem. The results show that allowing selection among the top three significantly increases the success probability and shifts the optimal stopping threshold earlier than in the classical problem. This model provides insight into realistic decision-making scenarios where top interviewees are more or less similar.

翻译：我们考虑了经典秘书问题的两种变体。 * 一种为具有返回机制的秘书问题变体，其中每位面试者可能以固定概率 p 第二次出现。决策者按顺序观察面试者，必须决定接受或拒绝每次出现。我们刻画了最优阈值规则，并考察了其与重现身概率 p 的依赖关系，突出了重复出现提供的额外信息如何改进选择性能。 * 另一种秘书问题变体将成功定义为选择三位最佳面试者中的任意一位（而非仅选最优）。面试者以随机顺序依次出现，且决策不可撤销。我们使用经典秘书问题的阈值策略估算了该放松成功标准下的成功概率。结果表明，允许在三位最佳面试者中进行选择会显著提升成功概率，且使最优停止阈值较经典问题提前。该模型揭示了顶级面试者之间差异不大的现实决策情境。