Seminars and Events - Research Colloquia

Probability Seminar

Department of Statistics
and
Department of Mathematics

Monday, September 23, 2002
3:30 PM in REC 113

Professor Stephen M. Samuels
Purdue University

will speak on

Two [or three or four] Quite Different Best-Choice Problems are Surprisingly Similar

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, replaces 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 problem and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery begs the question: Why?

I have a good explanation of the equivalence of the optimal rules. But so far, even under the lens of a planar Poisson process model, the equivalence of the asymptotic risks remains a mystery. Meanwhile two other problems have been shown to have the same limiting risks: the full-information problem with the (sub-optimal) Porosiński-Petruccelli stopping rule, and the Ferguson-Hardwick-Tamaki ``duration of holding the best'' problem, which turns out to be nothing but the Porosiński problem in disguise.