Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T21:24:10.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal multivariate stopping rules

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

David Assaf  and
Ester Samuel-Cahn
David Assaf*
Affiliation:
The Hebrew University of Jerusalem
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Department of Statistics, Hebrew University, Jerusalem 91905, Israel.
∗Postal address: Department of Statistics, Hebrew University, Jerusalem 91905, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

For fixed i let X(i) = (X1(i), …, Xd(i)) be a d-dimensional random vector with some known joint distribution. Here i should be considered a time variable. Let X(i), i = 1, …, n be a sequence of n independent vectors, where n is the total horizon. In many examples Xj(i) can be thought of as the return to partner j, when there are d ≥ 2 partners, and one stops with the ith observation. If the jth partner alone could decide on a (random) stopping rule t, his goal would be to maximize EXj(t) over all possible stopping rules t ≤ n. In the present ‘multivariate’ setup the d partners must however cooperate and stop at the same stopping time t, so as to maximize some agreed function h(∙) of the individual expected returns. The goal is thus to find a stopping rule t* for which h(EX1 (t), …, EXd(t)) = h (EX(t)) is maximized. For continuous and monotone h we describe the class of optimal stopping rules t*. With some additional symmetry assumptions we show that the optimal rule is one which (also) maximizes EZt where Zi = ∑dj=1Xj(i), and hence has a particularly simple structure. Examples are included, and the results are extended both to the infinite horizon case and to the case when X(1), …, X(n) are dependent. Asymptotic comparisons between the present problem of finding suph(EX(t)) and the ‘classical’ problem of finding supEh(X(t)) are given. Comparisons between the optimal return to the statistician and to a ‘prophet’ are also included. In the present context a ‘prophet’ is someone who can base his (random) choice g on the full sequence X(1), …, X(n), with corresponding return suph(EX(g)).

Keywords

Cooperative stopping rulefinite and infinite horizonmaximization of functions of expected valueoptimal choice setprophet inequalities

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 693 - 706
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported by Grant no. 94–00186 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

References

Assaf, D., and Samuel-Cahn, E. (1998). Optimal cooperative stopping rules for maximization of the product of expected stopped values. Stat. Prob. Lett. 38, 89–99.Google Scholar
Bassan, B., and Scarsini, M. (1995). On the value of information in multi-agent decision theory. J. Math. Econ. 24, 557–576.Google Scholar
Chow, Y. S., Robbins, H., and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Glickman, H. (1999). Cooperative stopping rules in multivariate problems. , The Hebrew University of Jerusalem.Google Scholar
Hill, T. P., and Kertz, R. P. (1981). Ratio comparisons of supremum and stop rule expectations. Z. Wahrscheinlichkeitsth. 56, 283–285.Google Scholar
Hill, T. P., and Kertz, R. P. (1992). A survey of prophet inequalities in optimal stopping theory. Contemporary Mathematics 125, American Mathematical Society, Providence, RI, pp. 191–207.Google Scholar
Kennedy, D. P., and Kertz, R. P. (1991). The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. 19, 329–341.Google Scholar
Lehmann, E. L. (1991). Testing Statistical Hypotheses, 2nd edn. Wadsworth & Brooks / Cole, Pacific Grove, CA.Google Scholar
Mamer, J. W. (1987). Monotone stopping games. J. Appl. Prob. 24, 386–401.Google Scholar
Marshall, A. W., and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Yasuda, M., Nakagami, J., and Kurano, M. (1982). Multivariate stopping problems with a monotone rule. J. Operat. Res. Soc. Japan 25, 334–349.Google Scholar