Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T06:12:54.244Z Has data issue: false hasContentIssue false
  • English
  • Français

On the optimal stopping problems with monotone thresholds

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Mitsushi Tamaki
Mitsushi Tamaki*
Affiliation:
Aichi University
*
∗Postal address: Department of Business Administration, Aichi University, Nagoya Campus, Hiraike 4–60–6, Nakamura, Nagoya, Aichi, 453-8777, Japan. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As a class of optimal stopping problems with monotone thresholds, we define the candidate-choice problem (CCP) and derive two formulae for calculating its expected payoff. We apply the first formula to a particular CCP, i.e. the best-choice duration problem treated by Ferguson et al. (1992). The recall case is also examined as a comparison. We also derive the distribution of the stopping time of the CCP and find, as a by-product, that the best-choice problem has a remarkable feature in that the optimal probability of choosing the best is just the expected value of the (proportional) stopping time. The similarity between the best-choice duration problem and the best-choice problem with uniform freeze studied by Samuel-Cahn (1996) is recognized.

Keywords

Secretary problembest-choice problemduration problembest-choice duration problemcandidate-choice problemmonotone ruleplanar Poisson process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 926 - 940
Copyright
Copyright © Applied Probability Trust 2015 

References

Bruss, F. T. and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Process. Appl. 38, 267–278.Google Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
Ferguson, T. S. (2006). Optimal Stopping and Applications. Available at http://www.math.ucla.edu/~tom/Stopping/Contents.html.Google Scholar
Ferguson, T. S., Hardwick, J. P. and Tamaki, M. (1992). Maximizing the duration of owning a relatively best object. In Strategies for Sequential Search and Selection in Real Time (Contemp. Math. 125), American Mathematical Society, Providence, RI, pp. 37–57.Google Scholar
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
Gnedin, A. V. (1996). On the full information best-choice problem. J. Appl. Prob. 33, 678–687.Google Scholar
Gnedin, A. V. (2004). Best choice from the planar Poisson process. Stoch. Process. Appl. 111, 317–354.CrossRefGoogle Scholar
Gnedin, A. V. (2005). Objectives in the best-choice problems. Sequential Anal. 24, 177–188.Google Scholar
Kurushima, A. and Ano, K. (2009). Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations. J. Appl. Prob. 46, 402–414.Google Scholar
Mazalov, V. V. and Tamaki, M. (2006). An explicit formula for the optimal gain in the full-information problem of owning a relatively best object. J. Appl. Prob. 43, 87–101.CrossRefGoogle Scholar
Pearce, C. E. M., Szajowski, K. and Tamaki, M. (2012). Duration problem with multiple exchanges. Numer. Algebra Control Optimization 2, 333–355.Google Scholar
Porosinski, Z. (1987). The full-information best choice problem with a random number of observations. Stoch. Process. Appl. 24, 293–307.Google Scholar
Sakaguchi, M. (1973). A note on the dowry problem. Rep. Statist. Appl. Res. Un. Japan. Sci. Eng. 20, 11–17.Google Scholar
Samuel-Cahn, E. (1996). Optimal stopping with random horizon with application to the full-information best-choice problem with random freeze. J. Amer. Statist. Assoc. 91, 357–364.Google Scholar
Samuels, S. M. (1982). Exact solutions for the full information best choice problem. Tech. Rep. 82–17, Department of Statistics, Purdue University.Google Scholar
Samuels, S. M. (2004). Why do these quite different best-choice problems have the same solutions? Adv. Appl. Prob. 36, 398–416.Google Scholar
Tamaki, M. (2009). Optimal choice of the best available applicant in full-information models. J. Appl. Prob. 46, 1086–1099.Google Scholar
Tamaki, M. (2010). Sum the multiplicative odds to one and stop. J. Appl. Prob. 47, 761–777.Google Scholar