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A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Tomomi Matsui  and
Katsunori Ano
Tomomi Matsui*
Affiliation:
Tokyo Institute of Technology
Katsunori Ano*
Affiliation:
Shibaura Institute of Technology
*
∗ Postal address: Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8550, Japan. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Shibaura Institute of Technology, Fukasaku, Minuma-ku, Saitama-shi, Saitama, 337-8570, Japan.
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Abstract

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In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

Keywords

Optimal stoppingodd problemlower boundsecretary problemMaclaurin's inequality

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 885 - 889
Copyright
© Applied Probability Trust 

References

Bruss, F. T. (2000). Sum the odds to one and stop. Ann. Prob. 28, 1384–1391.Google Scholar
Bruss, F. T. (2003). A note on bounds for the odds theorem of optimal stopping. Ann. Prob. 31, 1859–1861.CrossRefGoogle Scholar
Maclaurin, C. (1729). A second letter from Mr. Colin Maclaurin to Martin Folkes, Esq; Concerning the roots of equations, with the demonstration of other rules in algebra. Philos. Trans. 36, 59–96.Google Scholar
Tamaki, M. (2010). Sum the multiplicative odds to one and stop. J. Appl. Prob. 47, 761–777.CrossRefGoogle Scholar