Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T05:08:18.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimizing the expected rank with full information

Part of: Stochastic processes Sequential methods

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: [email protected]
∗∗ Postal address: University of California, Department of Mathematics, Los Angeles, CA 90024, USA. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

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.)

References

Assaf, D. and Samuel-Cahn, E. (1991) The secretary problem: minimizing the expected rank with i.i.d. random variables.Google Scholar
Chow, Y. S., Moriguti, S., Robbins, H. and Samuels, S. M. (1964) Optimal selection based on relative rank (the secretary problem). Israel J. Math. 2, 81–90.Google Scholar
Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
Kennedy, D. P. and Kertz, R. P. (1990) Limit theorems for threshold-stopped random variables with applications to optimal stopping. Adv. Appl. Prob. 22, 396–411.Google Scholar
Moser, L. (1956) On a problem of Cayley. Scripta Math. 22, 289–292.Google Scholar
Lindley, D. V. (1961) Dynamic programming and decision theory. Appl. Statist. 10, 39–51.Google Scholar