J. Appl. Prob. 43, 16–31 (2006) Printed in Israel  Applied Probability Trust 2006 RANGE OF ASYMPTOTIC BEHAVIOUR OF THE OPTIMALITY PROBABILITY OF THE EXPERT AND MAJORITY RULES DANIEL BEREND ∗ and LUBA SAPIR, ∗∗ Ben-Gurion University Abstract We study the uncertain dichotomous choice model. In this model, a group of expert decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision, and is optimal if it maximizes the probability of the group making a correct choice. In this paper, we assume the correctness probabilities of the experts to be independent random variables selected from some given distribution. Moreover, the ranking of the members in the group is (at least partly) known. Thus, one can follow rules based on this ranking. The extremes are the expert rule and the majority rule. The probabilities of the two extreme rules being optimal were compared in a series of early papers, for a variety of distributions. In most cases, the asymptotic behaviours of the probabilities of the two extreme rules followed the same patterns. Do these patterns hold in general? If not, what are the ranges of possible asymptotic behaviours of the probabilities of the two extreme rules being optimal? In this paper, we provide satisfactory answers to these questions. Keywords: Dichotomous choice; decision rule; expert rule; majority rule; optimality probability; partial information 2000 Mathematics Subject Classification: Primary 91B06 Secondary 91B12; 90B50 1. Introduction 1.1. Background There are many situations where a group of expert decision makers is required to select one of two alternatives, of which exactly one is regarded as correct. A decision rule is a rule for translating the individual opinions of the members into a group decision. The decision skill of each expert is characterized by the individual’s probability of making the right choice. There are several aspects to the study of this so-called dichotomous choice model. One of them is dealing with the Condorcet jury theorem in various setups. Condorcet [14] believed that a group of individuals facing a binary choice and using a simple majority rule is likely to make the correct choice. Moreover, this likelihood tends to complete certainty as the number of members of the group tends to infinity (see [2]). A Condorcet jury theorem is a formulation of conditions substantiating this belief. The classical conditions of this theorem assume the Received 22 February 2005; revision received 8 September 2005. ∗ Postal address: Departments of Mathematics and of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel. Email address: berend@cs.bgu.ac.il ∗∗ Postal address: Department of Industrial Engineering and Management, Ben-Gurion University, Beer-Sheva, 84105, Israel. Email address: lsapir@bgu.ac.il 16 at https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1143936240 Downloaded from https://www.cambridge.org/core. IP address: 34.228.24.229, on 21 May 2020 at 21:55:26, subject to the Cambridge Core terms of use, available