Judgment aggregation without full rationality

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinstein’s group identification problem. This paper was circulated in August 2006 and presented at the Yale workshop on Aggregation of Opinions, September 2006, at the Centre interuniversitaire de rechere en économie quantitative, Montreal, October 2006, and at the 1st International Workshop on Computational Social Choice, Amsterdam, December 2006. We are grateful to the participants at these occasions and the anonymous referees for comments. We also thank Ton Storcken for discussion. Elad Dokow and Ron Holzman have independently proved closely related results, which were also presented at the Yale workshop in September 2006, and circulated in the December 2006 paper (Dokow and Holzman 2006).