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We study a generalization of the secretary problem, where decisions do not have to be made immediately upon applicants’ arrivals. After arriving, each applicant stays in the system for some (random) amount of time and then leaves, whereupon the algorithm has to decide irrevocably whether to select this applicant or not. The arrival and waiting times are drawn from known distributions, and the decision maker’s goal is to maximize the probability of selecting the best applicant overall. Our first main result is a characterization of the optimal policy for this setting. We show that when deciding whether to select an applicant, it suffices to know only the time and the number of applicants that have arrived so far. Furthermore, the policy is monotone nondecreasing in the number of applicants seen so far, and, under certain natural conditions, monotone nonincreasing in time. Our second main result is that when the number of applicants is large, a single threshold policy is almost optimal. Funding: A. Psomas is supported in part by the National Science Foundation [Grant CCF-2144208], a Google Research Scholar Award, and by the Algorand Centres of Excellence program managed by Algorand Foundation. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.2476 .more » « less
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A single homogeneous resource needs to be fairly shared between users that dynamically arrive and depart over time. Although good allocations exist for any fixed number of users, implementing these allocations dynamically is impractical: it typically entails adjustments in the allocation of every user in the system whenever a new user arrives. We introduce a dynamic fair resource division problem in which there is a limit on the number of users that can be disrupted when a new user arrives and study the trade-off between fairness and the number of allowed disruptions, using a fairness metric: the fairness ratio. We almost completely characterize this trade-off and give an algorithm for obtaining the optimal fairness for any number of allowed disruptions.more » « less