A Simple Search Problem

Abstract

A simple problem is studied in which there are N boxes and a prize known to be in one of the boxes. Furthermore, the probability that the prize is in any box is given. It is desired to find the prize with minimal expected work, where it takes one unit of work to open a box and look inside. The paper establishes bounds on the minimal work in terms of the $p=1/2$ H\"older norm of the probability density and in terms of the entropy of the probability density. We also introduce the notion of "Cartesian product" of problems, and determine the asymptotic behavior of the minimal work for the $n$th power of a problem. (This article is a newly typeset version of an internal publication written in 1984. The second author passed away on November 12, 2020, and his estate has approved the submission of this paper.)