Journal of Applied Probability

What is known about Robbins' Problem?

F. Thomas Bruss

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Let X1, X2,..., Xn be independent, identically distributed random variables, uniform on [0,1]. We observe the Xk 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.

Article information

J. Appl. Probab. Volume 42, Number 1 (2005), 108-120.

First available in Project Euclid: 9 March 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Full information optimal selection secretary problem half-prophet memoryless rule history dependence truncation embedding integral-differential equation


Bruss, F. Thomas. What is known about Robbins' Problem?. J. Appl. Probab. 42 (2005), no. 1, 108--120. doi:10.1239/jap/1110381374.

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