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What is Known About Robbins' Problem?

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Département de Mathématique, Université Libre de Bruxelles, CP 210, B-1050 Brussels, Belgium. Email address: tbruss@ulb.ac.be
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Abstract

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Let X 1, X 2,…, X n be independent, identically distributed random variables, uniform on [0,1]. We observe the X k sequentially and must stop on exactly one of them. No recollection of the preceding observations is permitted. What stopping rule τ minimizes the expected rank of the selected observation? This full-information expected-rank problem is known as Robbins' problem. The general solution is still unknown, and only some bounds are known for the limiting value as n tends to infinity. After a short discussion of the history and background of this problem, we summarize what is known. We then try to present, in an easily accessible form, what the author believes should be seen as the essence of the more difficult remaining questions. The aim of this article is to evoke interest in this problem and so, simply by viewing it from what are possibly new angles, to increase the probability that a reader may see what seems to evade probabilistic intuition.

Keywords

Full informationoptimal selectionsecretary problemhalf-prophetmemoryless rulehistory dependencetruncationembeddingintegral-differential equation

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 108 - 120
Copyright
© Applied Probability Trust 2005 

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