The Condorcet set: Majority voting over interconnected propositions

Judgement aggregation is a model of social choice in which the space of social alternatives is the set of consistent evaluations (‘views’) on a family of logically interconnected propositions, or yes/no-issues. Yet, simply complying with the majority opinion in each issue often yields a logically inconsistent collection of judgements. Thus, we consider the Condorcet set: the set of logically consistent views which agree with the majority on a maximal set of issues. The elements of this set are exactly those that can be obtained through sequential majority voting, according to which issues are sequentially decided by simple majority unless earlier choices logically force the opposite decision. We investigate the size and structure of the Condorcet set —and hence the properties of sequential majority voting —for several important classes of judgement aggregation problems. While the Condorcet set verifies McKelvey’s (1979) celebrated ‘chaos theorem’ in a number of contexts, in others it is shown to be very regular and well-behaved. In the context of preference aggregation, pairwise majority voting often fails to produce unambiguous outcomes because a Condorcet winner might not exist. Similarly, in the general judgement aggregation problem where each social outcome is described as a pattern ∗We are grateful to an editor and a referee for their helpful comments and suggestions on an earlier version of this work. We would also like to thank audiences at the Workshop of the Centre of Mathematical Social Sciences at the University of Auckland, Tagung des Theoretischen Ausschusses des Vereins für Socialpolitik in Günzburg, the OECD conference “New Directions in Welfare” in Paris, the Meeting of the Society for Social Choice and Welfare in New Delhi, the COMSOC conference in Krakow, and of seminars at Université de Strasbourg, Université de Lausanne, Universität Wien, Ludwig-Maximilians-Univerisität München, Yale University, SUNY Stony Brook, Universität Graz, Universität Innsbruck and GRECAM in Marseille for comments and remarks. †Dept. of Economics, University of California at Davis, [email protected]. Address: 1255 New Hampshire Ave NW, Apt. 825, Washington, DC 20036, U.S.A. Phone: (+1) 202 296 9360. ‡Dept. of Mathematics, Trent University, Canada, [email protected]. Address: 1600 West Bank Drive, Peterborough, Ontario, Canada K9J 7B8. Phone: (+1) 705 748 1011. §Dept.of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany, [email protected]. Address: Kaiserstrasse 12, 76128 Karlsruhe, Germany. Phone: (+49) 721 608 43077 (corresponding author).