EFFECTIVENESS OF WEIGHTED MAJORITY RULES WITH RANDOM DECISION POWER DISTRIBUTION DANIEL BEREND and YURI CHERNYAVSKY Ben-Gurion University Abstract There is a large body of research studying the conditions un- der which majority decisions are best. In particular, in many circumstances, the probability of a group to decide correctly is higher than that of a random subgroup. Moreover, the lat- ter probability increases as the subgroup size grows. Here we generalize those results by showing that, in the same setup, the simple majority rule is the most effective of all weighted majority rules when weights are distributed randomly be- tween group members. For special families of weighted ma- jority rules, rule effectiveness increases as we get “closer” to the simple majority rule. 1. Introduction Consider a group of experts that has to collectively choose between two al- ternatives, exactly one of which is correct. Suppose that the alternatives are equally likely to be correct and the benefit of selecting the correct alternative is independent of the identity of this alternative. This model is known as the dichotomous choice model . Each member of the committee selects independently and has his own probability of selecting the correct alternative, indicating his competence level. A decision rule translates the individual opinions of the group members into a group decision. An example is the simple majority rule (defined for an odd group size), which selects the alternative chosen by more than half of the committee members. The opposite rule is the expert rule, that takes into Daniel Berend, Departments of Mathematics and of Computer Science, Ben-Gurion Uni- versity, Beer-Sheva 84105, Israel (berend@cs.bgu.ac.il). Yuri Chernyavsky, Department of Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel (yurac@cs.bgu.ac.il). The authors would like to express their gratitude to Ruth Ben-Yashar for her helpful comments on the paper. Received December 29, 2006; Accepted September 24, 2007. C  2008 Blackwell Publishing, Inc. Journal of Public Economic Theory, 10 (3), 2008, pp. 423–439. 423