Partitioning a sample using binary-type questions with ternary feedback

Abstract

The problem is to find the largest observation in a random sample of size n by asking binary-type questions of people (or items) in the sample. At each stage of the search, a threshold is calculated and a binary-type question is posed to each member of the sample. The threshold is determined from answers given to the previous questions, and no exact data is ever collected, i.e., no member is asked to explicitly provide his observation. Arrow, Pesotchinsky, and Sobel (APS) calculated the optimal threshold sequence for two different objectives: (i) minimize the average number of questions required for a solution, and (ii) maximize the probability of solving the problem in, at most, r questions. APS have assumed that the number of respondents in the affirmative at each stage of the search is exactly known. There exist applications where the number of affirmative answers is only known to be one member of the set (0, 1, more than 1). For these applications, we calculate exactly the optimal thresholds, in the sense of maximizing the probability of getting precisely one affirmative answer to the next binary question. An application of the threshold-calculation procedure is demonstrated in resolution of packet collisions over multiuser communication channel.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>