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Minimizing the expected rank with full information

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss  and
Thomas S. Ferguson
F. Thomas Bruss*
Affiliation:
Vrije Universiteit Brussel–Vesalius College
Thomas S. Ferguson*
Affiliation:
University of California at Los Angeles
*
∗ Postal address: Vrije Universiteit Brussel, Department of Mathematics, B-1050 Brussels, Belgium. E-mail: ftbruss@tena2.vub.ac.be
∗∗ Postal address: University of California, Department of Mathematics, Los Angeles, CA 90024, USA. E-mail: tom@math.ucla.edu
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Abstract

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.

Keywords

SECRETARY PROBLEMOPTIMAL STOPPING

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 616 - 626
Copyright
Copyright © Applied Probability Trust 1993 

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References

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