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The optimal value of markov stopping problems with one-step look ahead policy

Published online by Cambridge University Press:  14 July 2016

Masami Yasuda
Masami Yasuda*
Affiliation:
Chiba University
*
∗ Postal address: College of General Education, Chiba University, Chiba, 260 Japan.
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Abstract

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.

Keywords

OPTIMAL STOPPING PROBLEMMARKOV POTENTIAL THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 3 , September 1988 , pp. 544 - 552
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Bojdecki, T. (1978) On optimal stopping of a sequence of independent random variables—Probability maximizing approach. Stoch. Proc. Appl. 6, 153–163.Google Scholar
[2] Chow, Y. S., Robbins, H. and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
[3] Cowan, R. and Zabczyk, J. (1978) An optimal selection problem associated with the Poisson process. Theory Prob. Appl. 23, 584–592.Google Scholar
[4] Darling, D. A. (1985) Contribution to the optimal stopping problem, Z. Wahrscheinlichkeitsth. 70, 525–533.Google Scholar
[5] Dubins, L. E. and Savage, L. J. (1965) How to Gamble if You Must: Inequalities for Stochastic Processes. McGraw-Hill, New York.Google Scholar
[7] Dynkin, E. B. and Yushkevitch, A. A. (1969) Theorems and Problems on Markov Processes. Plenum Press, New York.Google Scholar
[8] Freeman, P. R. (1983) The secretary problem and its extensions: a review. Int. Statist. Rev. 51, 189–206.Google Scholar
[9] Gianini, J. P. and Samuels, S. M. (1976) The infinite secretary problem. Ann. Prob. 13, 418–432.Google Scholar
[10] Hordijk, A. (1974) Dynamic Programming and Markov Potential Theory. Mathematisch Centrum, Amsterdam.Google Scholar
[11] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976) Denumerable Markov Chains, 2nd edn. Springer-Verlag, Berlin.Google Scholar
[12] Mucci, A. G. (1973) Differential equations and optimal choice problem. Ann. Statist. 1, 104–113.Google Scholar
[13] Presman, E. L. and Sonin, I. M. (1972) The best choice problem for a random number of objects. Theory Prob. Appl. 17, 657–668.Google Scholar
[14] Rasmussen, W. T. and Robbins, H. (1975) The candidate problem with unknown population size. J. Appl. Prob. 12, 692–701.Google Scholar
[15] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden Day, San Francisco.Google Scholar
[16] Smith, M. H. (1975) A secretary problem with uncertain employment. J. Appl. Prob. 12, 620–624.Google Scholar
[17] Yasuda, M. (1984) Asymptotic results for the best-choice problem with a random number of objects. J. Appl. Prob. 21, 521–536.Google Scholar