IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i2p194-d720597.html
   My bibliography  Save this article

On Asymptotics of Optimal Stopping Times

Author

Listed:
  • Hugh N. Entwistle

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

  • Christopher J. Lustri

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

  • Georgy Yu. Sofronov

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

Abstract

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Suggested Citation

  • Hugh N. Entwistle & Christopher J. Lustri & Georgy Yu. Sofronov, 2022. "On Asymptotics of Optimal Stopping Times," Mathematics, MDPI, vol. 10(2), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:194-:d:720597
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/2/194/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/2/194/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Stein, William E. & Seale, Darryl A. & Rapoport, Amnon, 2003. "Analysis of heuristic solutions to the best choice problem," European Journal of Operational Research, Elsevier, vol. 151(1), pages 140-152, November.
    2. Georgy Sofronov, 2020. "An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers," Mathematics, MDPI, vol. 8(5), pages 1-11, May.
    3. David, Israel, 1998. "Explicit results for a class of asset-selling problems," European Journal of Operational Research, Elsevier, vol. 110(3), pages 576-584, November.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. J. Neil Bearden & Amnon Rapoport & Ryan O. Murphy, 2006. "Sequential Observation and Selection with Rank-Dependent Payoffs: An Experimental Study," Management Science, INFORMS, vol. 52(9), pages 1437-1449, September.
    2. Georgy Sofronov, 2020. "An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers," Mathematics, MDPI, vol. 8(5), pages 1-11, May.
    3. Darryl Seale & William Stein & Amnon Rapoport, 2014. "Hold or roll: reaching the goal in jeopardy race games," Theory and Decision, Springer, vol. 76(3), pages 419-450, March.
    4. Sofronov, Georgy, 2013. "An optimal sequential procedure for a multiple selling problem with independent observations," European Journal of Operational Research, Elsevier, vol. 225(2), pages 332-336.
    5. Ramsey, David M. & Szajowski, Krzysztof, 2004. "Correlated equilibria in competitive staff selection problem," MPRA Paper 19870, University Library of Munich, Germany, revised 2006.
    6. Georgy Yu. Sofronov, 2020. "An Optimal Double Stopping Rule for a Buying-Selling Problem," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 1-12, March.
    7. Vincent Mak & Darryl A. Seale & Amnon Rapoport & Eyran J. Gisches, 2019. "Voting Rules in Sequential Search by Committees: Theory and Experiments," Management Science, INFORMS, vol. 65(9), pages 4349-4364, September.
    8. Andrej Angelovski & Werner Güth, 2017. "When to Stop - A Cardinal Secretary Search Experiment," Working Papers CESARE 1706, Dipartimento di Economia e Finanza, LUISS Guido Carli.
    9. Mak, Vincent & Rapoport, Amnon & Seale, Darryl A., 2014. "Sequential search by groups with rank-dependent payoffs: An experimental study," Organizational Behavior and Human Decision Processes, Elsevier, vol. 124(2), pages 256-267.
    10. Jhunjhunwala, Tanushree, 2021. "Searching to avoid regret: An experimental evidence," Journal of Economic Behavior & Organization, Elsevier, vol. 189(C), pages 298-319.
    11. Sugden, Robert & Wang, Mengjie & Zizzo, Daniel John, 2019. "Take it or leave it: Experimental evidence on the effect of time-limited offers on consumer behaviour," Journal of Economic Behavior & Organization, Elsevier, vol. 168(C), pages 1-23.
    12. Rami Zwick & Amnon Rapoport & Alison King Chung Lo & A. V. Muthukrishnan, 2003. "Consumer Sequential Search: Not Enough or Too Much?," Marketing Science, INFORMS, vol. 22(4), pages 503-519, October.
    13. Kausel, Edgar E. & Culbertson, Satoris S. & Madrid, Hector P., 2016. "Overconfidence in personnel selection: When and why unstructured interview information can hurt hiring decisions," Organizational Behavior and Human Decision Processes, Elsevier, vol. 137(C), pages 27-44.
    14. Ismail Saglam, 2014. "Simple Heuristics as Equilibrium Strategies in Mutual Sequential Mate Search," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 17(1), pages 1-12.
    15. Shuk Ying Ho & David Bodoff & Kar Yan Tam, 2011. "Timing of Adaptive Web Personalization and Its Effects on Online Consumer Behavior," Information Systems Research, INFORMS, vol. 22(3), pages 660-679, September.
    16. Didrika S. van de Wouw & Ryan T. McKay & Bruno B. Averbeck & Nicholas Furl, 2022. "Explaining human sampling rates across different decision domains," Judgment and Decision Making, Society for Judgment and Decision Making, vol. 17(3), pages 487-512, May.
    17. Demers, Simon, 2021. "Expected duration of the no-information minimum rank problem," Statistics & Probability Letters, Elsevier, vol. 168(C).
    18. Amnon Rapoport & Darryl A. Seale & Leonidas Spiliopoulos, 2023. "Progressive stopping heuristics that excel in individual and competitive sequential search," Theory and Decision, Springer, vol. 94(1), pages 135-165, January.
    19. Schunk, Daniel & Winter, Joachim, 2009. "The relationship between risk attitudes and heuristics in search tasks: A laboratory experiment," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 347-360, August.
    20. Stephan Billinger & Kannan Srikanth & Nils Stieglitz & Terry R. Schumacher, 2021. "Exploration and exploitation in complex search tasks: How feedback influences whether and where human agents search," Strategic Management Journal, Wiley Blackwell, vol. 42(2), pages 361-385, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:194-:d:720597. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.