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Why do these quite different best-choice problems have the same solutions?

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2016

Stephen M. Samuels
Stephen M. Samuels*
Affiliation:
Purdue University
*
∗ Postal address: Department of Statistics, Mathematical Sciences Building, 150 North University Street, West Lafayette, IN 47907-2067, USA. Email address: ssamuels@stat.purdue.edu
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Abstract

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

Keywords

Secretary problemoptimal stoppingplanar Poisson process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 398 - 416
Copyright
Copyright © Applied Probability Trust 2004 

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