A geometric approach to judgement aggregation

The problem of judgement aggregation consists in aggregating individual judgments on an agenda of logically interconnected propositions into a collective set of judgments on these propositions. This relatively new literature (see List and Puppe (2007) for a survey) is centred on problems like the discursive dilemma which are structurally similar to paradoxes and problems in social choice theory like the Condorcet paradox and Arrow’s general possibility theorem. Saari (1995) has successfully introduced a geometric approach to the analysis of such paradoxes the extension of which to judgment aggregation seems promising. A major difference of judgement aggregation to social choice theory lies in the representation of the information involved. While binary relations over a set of alternatives are a natural representation of preferences, judgments are typically represented by sets of propositions or by vectors of their valuations, where the logical interconnections between these propositions determine the set of feasible valuations. E.g. the agenda (p, q, p∧ q) is associated the set of feasible valuations {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1)} . In this paper we want to use Saari’s tools to analyse results in judgement aggregation. In particular we will first use Saari’s representation cubes to provide a geometric presentation of profiles and majority rule outcomes. Applying Saari’s idea of a profile decomposition we will show what can go wrong in certain domains of judgment aggregation and how problems can be avoided with the help of domain restrictions. Moreover, we will show that usual qualified majorities can not resolve such paradoxical situations.