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A Random Arrival Time Best-Choice Problem with Uniform Prior on the Number of Arrivals

  • Mitsushi TamakiEmail author
  • Qi Wang
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 39)

Summary

Suppose that a random number N of rankable applicants appear and their arrival times are i.i.d. random variables having a known distribution function. A method of choosing the best applicant is investigated when a prior on N is uniform on \(\{1,2,\ldots ,n\}\). An exact form of the optimal selection rule is derived. Stewart first studied this problem, but examined only the case of the non-informative prior, i.e., the limiting case of \(n\to \infty\), so our result can be considered as a generalization of Stewart’s result.

Keywords

secretary problem optimal stopping bayesian updating OLA rule \(e^{-1}\)-rule relative rank 

Notes

Acknowledgment

We are grateful to the anonymous referees for their careful reading.

References

  1. 1.
    Bojdecki, T.: On optimal stopping of a sequence of independent random variables - probability maximizing approach. Stoch. Proc. Their Appl. 6, 153–163 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bruss, F.T.: A unified approach to a class of best choice problems with an unknown number of objects. Ann. Probab. 12, 882–889 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bruss, F.T.: On an optimal selection problem of Cowan and Zabczyk. J. Appl. Probab. 24, 918–928 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bruss, F.T., Samuels, S.M.: A unified approach to a class of optimal selection problems with an unknown number of options. Ann. Probab. 15, 824–830 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bruss, F.T., Samuels, S.M.: Conditions for quasi-stationarity of the Bayes rule in selection problems with an unknown number of rankable options. Ann. Probab. 18, 877–886 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bruss, F.T., Rogers, L.C.G.: Pascal processes and their characterization. Stoch. Proc. Their Appl. 37, 331–338 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chow, Y., Robbins, H., Siegmund, D.: The Theory of Optimal Stopping, Houghton Mifflin, Boston, MA (1971)zbMATHGoogle Scholar
  8. 8.
    Cowan, R., Zabczyk, J.: An optimal selection problem associated with the poisson process. Theory Probab. Appl. 23, 584–592 (1978)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gianini, J., Samuels, S.M.: The infinite secretary problem. Ann. Probab. 4, 418–432 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gilbert, J.P., Mosteller, F.: Recognizing the maximum of a sequence. J. Am. Stat. Assoc. 61, 35–73 (1966)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Presman, E.L., Sonin, I.M.: The best choice problem for a random number of objects. Theory Probab. Appl. 17, 657–668 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ross, S.M.: Introduction to Stochastic Dynamic Programming, Academic, Orland, CA (1983)zbMATHGoogle Scholar
  13. 13.
    Sakaguchi, M.: Optimal stopping problems for randomly arriving offers. Math. Japon. 21, 201–217 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Sakaguchi, M., Tamaki, M.: Optimal stopping problems associated with a nonhomogeneous Markov process. Math. Japon. 25, 681–696 (1980)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Stewart, T.J.: The secretary problem with an unknown number of options.Opns. Res. 29, 130–145 (1981)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Business AdministrationAichi UniversityAichiJapan

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