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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

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