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Odds Theorem with Multiple Selection Chances

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Katsunori Ano ,
Hideo Kakinuma  and
Naoto Miyoshi
Katsunori Ano*
Affiliation:
Institute of Applied Mathematics
Hideo Kakinuma*
Affiliation:
Tokyo Institute of Technology
Naoto Miyoshi*
Affiliation:
Tokyo Institute of Technology
*
∗Postal address: Department of Applied Probability, Institute of Applied Mathematics, Asakusabashi, Taito-ku, Tokyo 111-0053, Japan. Email address: kano@iapm.jp
∗∗Postal address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-52 Ookayama, Meguro-ku, Tokyo 152-8552, Japan.
∗∗∗Postal address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-52 Ookayama, Meguro-ku, Tokyo 152-8552, Japan. Email address: miyoshi@is.titech.ac.jp
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Abstract

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We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

Keywords

Optimal stoppingselecting the last successmultiple selection chances

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1093 - 1104
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Ano, K. and Ando, A. (2000). A note on Bruss' stopping problem with random availability. In Game Theory, Optimal Stopping, Probability and Statistics (IMS Lecture Notes Monogr. 35), Institute for Mathematical Statistics, Beachwood, OH, pp. 71–82.Google Scholar
[2] Bruss, F. T. (1988). Invariant record processes and applications to best choice modelling. Stoch. Process. Appl. 30, 303–316.Google Scholar
[3] Bruss, F. T. (2000). Sum the odds to one and stop. Ann. Prob. 28, 1384–1391.Google Scholar
[4] Bruss, F. T. (2003). A note on bounds for the odds theorem of optimal stopping. Ann. Prob. 31, 1859–1861.Google Scholar
[5] Bruss., F. T. and Paindaveine, D. (2000). Selecting a sequence of last successes in independent trials. J. Appl. Prob. 37, 389–399.CrossRefGoogle Scholar
[6] Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
[7] Ferguson, T. S. (2006). Optimal Stopping and Applications. Available at http://www.math.ucla.edu/∼tom/Stopping/Contents.html.Google Scholar
[8] Ferguson, T. S. (2008). The sum-the-odds theorem with application to a stopping game of Sakaguchi. Preprint.Google Scholar
[9] Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
[10] Hill, T. P. and Krengel, U. (1992). A prophet inequality related to the secretary problem. In (Contemp. Math. 125), eds Bruss, F. T., Ferguson, T. S. and Samuels, S. M., American Mathematical Society, Providence, RI, pp. 209–215.Google Scholar
[11] Hsiau, S.-R. and Yang, J.-R. (2000). A natural variation of the standard secretary problem. Statist. Sinica 10, 639–646.Google Scholar
[12] Hsiau, S.-R. and Yang, J.-R. (2002). Selecting the last success in Markov-dependent trials. J. Appl. Prob. 39, 271–281.Google Scholar
[13] Sakaguchi, M. (1978). Dowry problems and OLA policies. Rep. Statist. Appl. Res. Union Japan. Sci. Eng. 25, 24–28.Google Scholar
[14] Sakaguchi, M. (1984). Bilateral sequential games related to the no-information secretary problem. Math. Japon. 29, 961–973.Google Scholar