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This paper explores a modern variation of the classical secretary problem, also known as the marriage or best-choice problem. In this “21st-century” adaptation, a decision-maker sequentially evaluates \(n \in \mathbb {N}\) uniquely ranked candidates, where the ranks range from 1 (the worst) to n (the best), with no ties. The strategy begins with a non-selection phase, during which the first \(d \in \mathbb {N}\) candidates are observed but not selected, with \(d < n\) . After this phase, the decision-maker considers selecting the first subsequent candidate who is better than all those seen so far. Crucially, upon encountering such a candidate, the decision-maker invokes a lookahead window of size \(l \in \mathbb {N}\) , meaning they evaluate not only the current candidate—who is the best so far—but also the next \(l - 1\) candidates (up to a maximum of n candidates), assuming \(l \le n\) . This mechanism models the modern ability to quickly contact and assess a small group of potential candidates, a feature enabled by contemporary communication technologies and the inspiration for calling this the “21st-century” version of the problem. From this lookahead group, the candidate with the highest rank is selected. A selection is considered successful if and only if the chosen candidate is the overall best among all n. The main contribution of this work is the analytical derivation and numerical analysis of the probability P(d, n, l), which denotes the probability that the selected candidate is the best overall under this lookahead-enhanced strategy. We develop a continuous approximation of P(d, n, l) using harmonic series and analyze its maximization to determine the optimal stopping fraction \(\frac{d}{n}\) that maximizes the success probability. Our findings generalize the classical optimal threshold \(\frac{d}{n} \approx e^{-1}\) by incorporating the effect of the lookahead window l. Assuming that \(l \ll n\) , we show that the optimal stopping threshold becomes \(\frac{d}{n} \approx e^{-1 - \frac{l - 1}{n}}\) , which identifies the point in the process where the decision-maker should transition from the non-selection to the selection phase. At this threshold, the maximum success probability is approximately \(P(d, n, l) \approx e^{-1 - \frac{l - 1}{n}} + \frac{l - 1}{n}\) . This lookahead mechanism reflects realistic selection scenarios where limited short-term planning and rapid evaluation—enabled by modern technologies—can enhance decision-making outcomes. The results provide both theoretical insight and practical guidance for optimal stopping strategies in dynamic, ranked-choice environments where only one selection is allowed and the goal is to secure the best possible candidate.

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The Secretary Problem for the 21st Century: Optimal One-Try Selection of the Best Candidate with a Lookahead Strategy

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  • 发表日期：

    2026年

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  • Lubomír Štěpánek

摘要

This paper explores a modern variation of the classical secretary problem, also known as the marriage or best-choice problem. In this “21st-century” adaptation, a decision-maker sequentially evaluates \(n \in \mathbb {N}\) uniquely ranked candidates, where the ranks range from 1 (the worst) to n (the best), with no ties. The strategy begins with a non-selection phase, during which the first \(d \in \mathbb {N}\) candidates are observed but not selected, with \(d < n\) . After this phase, the decision-maker considers selecting the first subsequent candidate who is better than all those seen so far. Crucially, upon encountering such a candidate, the decision-maker invokes a lookahead window of size \(l \in \mathbb {N}\) , meaning they evaluate not only the current candidate—who is the best so far—but also the next \(l - 1\) candidates (up to a maximum of n candidates), assuming \(l \le n\) . This mechanism models the modern ability to quickly contact and assess a small group of potential candidates, a feature enabled by contemporary communication technologies and the inspiration for calling this the “21st-century” version of the problem. From this lookahead group, the candidate with the highest rank is selected. A selection is considered successful if and only if the chosen candidate is the overall best among all n. The main contribution of this work is the analytical derivation and numerical analysis of the probability P(d, n, l), which denotes the probability that the selected candidate is the best overall under this lookahead-enhanced strategy. We develop a continuous approximation of P(d, n, l) using harmonic series and analyze its maximization to determine the optimal stopping fraction \(\frac{d}{n}\) that maximizes the success probability. Our findings generalize the classical optimal threshold \(\frac{d}{n} \approx e^{-1}\) by incorporating the effect of the lookahead window l. Assuming that \(l \ll n\) , we show that the optimal stopping threshold becomes \(\frac{d}{n} \approx e^{-1 - \frac{l - 1}{n}}\) , which identifies the point in the process where the decision-maker should transition from the non-selection to the selection phase. At this threshold, the maximum success probability is approximately \(P(d, n, l) \approx e^{-1 - \frac{l - 1}{n}} + \frac{l - 1}{n}\) . This lookahead mechanism reflects realistic selection scenarios where limited short-term planning and rapid evaluation—enabled by modern technologies—can enhance decision-making outcomes. The results provide both theoretical insight and practical guidance for optimal stopping strategies in dynamic, ranked-choice environments where only one selection is allowed and the goal is to secure the best possible candidate.