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The Secretary Problem with an Unknown Number of Options

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  • T. J. Stewart

    (National Research Institute for Mathematical Sciences, Pretoria, South Africa)

Abstract

A method of selecting the best element from a random sequence of unknown length is investigated. By assuming that the arrival times of the elements are independent identically distributed (i.i.d.) exponential random variables, a procedure is established that maximizes the probability of selecting the best element. Asymptotically for large values of the actual length of the sequence, the optimal probability is 1/ e , which is also the corresponding asymptotic optimal value when the length is known. It is shown that the method behaves well even when the actual number of options is comparatively small, and that it is not particularly sensitive to errors in the specification of the arrival rate of the process.

Suggested Citation

  • T. J. Stewart, 1981. "The Secretary Problem with an Unknown Number of Options," Operations Research, INFORMS, vol. 29(1), pages 130-145, February.
    • Handle: RePEc:inm:oropre:v:29:y:1981:i:1:p:130-145
    DOI: 10.1287/opre.29.1.130
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    Cited by:

    1. Das, Sanmay & Tsitsiklis, John N., 2010. "When is it important to know you've been rejected? A search problem with probabilistic appearance of offers," Journal of Economic Behavior & Organization, Elsevier, vol. 74(1-2), pages 104-122, May.
    2. Ferenstein, Elzbieta Z. & Krasnosielska, Anna, 2010. "No-information secretary problems with cardinal payoffs and Poisson arrivals," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 221-227, February.
    3. Bruss, F. Thomas & Rogers, L.C.G., 2022. "The 1/e-strategy is sub-optimal for the problem of best choice under no information," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1059-1067.
    4. Anton J. Kleywegt & Jason D. Papastavrou, 1998. "The Dynamic and Stochastic Knapsack Problem," Operations Research, INFORMS, vol. 46(1), pages 17-35, February.
    5. Anton J. Kleywegt & Jason D. Papastavrou, 2001. "The Dynamic and Stochastic Knapsack Problem with Random Sized Items," Operations Research, INFORMS, vol. 49(1), pages 26-41, February.
    6. Alexander Gnedin & Zakaria Derbazi, 2022. "Trapping the Ultimate Success," Mathematics, MDPI, vol. 10(1), pages 1-19, January.
    7. Seale, Darryl A. & Rapoport, Amnon, 1997. "Sequential Decision Making with Relative Ranks: An Experimental Investigation of the "Secretary Problem">," Organizational Behavior and Human Decision Processes, Elsevier, vol. 69(3), pages 221-236, March.
    8. Gnedin, Alexander, 2022. "The best choice problem with random arrivals: How to beat the 1/e-strategy," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 226-240.

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