The geometry of consistent majoritarian judgement aggregation

Given a set of propositions with unknown truth values, a ‘judgement aggregation rule’ is a way to aggregate the personal truth-valuations of a set of jurors into some ‘collective’ truth valuation. We introduce the class of ‘quasimajoritarian’ judgement aggregation rules, which includes majority vote, but also includes some rules which use different weighted voting schemes to decide the truth of different propositions. We show that if the profile of jurors’ beliefs satisfies a condition called ‘value restriction’, then the output of any quasimajoritarian rule is logically consistent; this directly generalizes the recent work of Dietrich and List (2007). We then provide two sufficient conditions for value-restriction, defined geometrically in terms of a lattice ordering or an ultrametric structure on the set of jurors and propositions. Finally, we introduce another sufficient condition for consistent majoritarian judgement aggregation, called ‘convexity’. We show that convexity is not logically related to value-restriction. Let P be a finite set of propositions and let J be a finite jury. For all j ∈ J , let Pj ⊂ P be j’s judgement set: the set of propositions which j believes are true. Assume Pj is logically consistent, for each j ∈ J . The list P := (Pj)j∈J is called a judgement profile. A judgement aggregation rule is a function R which converts any judgement profile P into an aggregate judgement set R(P) ⊂ P; heuristically, R(P) is the set of propositions which are judged to be ‘true’ by the jury J as a whole. For example, the simple majoritarian rule Rmaj works as follows: For all p ∈ P, let Jp := {j ∈ J ; p ∈ Pj}. Then define Rmaj(P) := {p ∈ P ; |Jp| > |J |/2}. The problem is that Rmaj(P) may be inconsistent; this phenomenon was called the Doctrinal Paradox by Kornhauser and Sager (1986, 1993) in the context of jurisprudence. List and Pettit (2002) called this phenomenon the Discursive Dilemma, and showed that it is inevitable using any ‘reasonable’ judgement aggregation rule (not just Rmaj). Since then, the Dilemma has been the subject of intense investigation; see List and Puppe (2007) for a survey. Dietrich and List (2007; Proposition 16) have shown that if the profile P satisfies a structural condition called value restriction, then Rmaj(P) will be consistent. Value restriction is a somewhat abstract property without any clear social or epistemological interpretation, but Dietrich and List also provide several geometrically appealing sufficient