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The best choice problem with random arrivals: How to beat the 1/e-strategy

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  • Gnedin, Alexander

Abstract

In the best choice problem with random arrivals, an unknown number n of rankable items arrive at times sampled from the uniform distribution. As is well known, a real-time player can ensure stopping at the overall best item with probability at least 1/e, by waiting until time 1/e then selecting the first relatively best item (record) to appear, if available. This paper discusses the issue of dominance in a wide class of multi-cutoff stopping strategies of best choice, and argues that in fact the player faces a trade-off between success probabilities for various values of n. We show that the 1/e-strategy is not a unique minimax strategy and that it can be improved in various ways.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:145:y:2022:i:c:p:226-240
    DOI: 10.1016/j.spa.2021.12.008
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    References listed on IDEAS

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    1. Mitsushi Tamaki & Qi Wang, 2010. "A Random Arrival Time Best-Choice Problem with Uniform Prior on the Number of Arrivals," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 499-510, Springer.
    2. Browne, Sid & Bunge, John, 1995. "Random record processes and state dependent thinning," Stochastic Processes and their Applications, Elsevier, vol. 55(1), pages 131-142, January.
    3. Bruss, F. Thomas & Rogers, L. C. G., 1991. "Embedding optimal selection problems in a Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 38(2), pages 267-278, August.
    4. T. J. Stewart, 1981. "The Secretary Problem with an Unknown Number of Options," Operations Research, INFORMS, vol. 29(1), pages 130-145, February.
    5. Bruss, F. Thomas, 1988. "Invariant record processes and applications to best choice modelling," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 303-316, December.
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