Abstract
In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, non-dictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or extended unanimity preservation. We fully characterize the agendas for which there are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (Strategy-proof social choice on single-peaked domains: possibility, impossibility and the space between, 2002), Nehring (Oligarchies in judgment aggregation: a characterization, 2006), Dietrich and List (Soc Choice Welf 29(1):19–33, 2007a) and Dokow and Holzman (J Econ Theory 145(2):495–511, 2010a).
Highlights
Many democratically organized groups, such as electorates, legislatures, committees, juries and expert panels, are faced with the problem of judgment aggregation: They have to make collective judgments on certain propositions on the basis of the group F
We prove new results on the existence of propositionwise aggregation rules which are non-oligarchic, anonymous, give no individual veto power, or are extended-unanimity-preserving, as defined below
Corollary 3 There exist propositionwise aggregation rules F : Un → U without individual veto power for all sufficiently large group sizes n if and only if the agenda is not minimally blocked. This corollary remains true if anonymity, monotonicity or unanimity preservation are added as conditions on F
Summary
Many democratically organized groups, such as electorates, legislatures, committees, juries and expert panels, are faced with the problem of judgment aggregation: They have to make collective judgments on certain propositions on the basis of the group. A much-cited result shows that, if (and only if) the decision problem under consideration, called the agenda, has two combinatorial properties, as defined below, the only judgment aggregation rules satisfying these conditions are the dictatorships (Dokow and Holzman 2010a; the ‘if’ part was independently obtained by Dietrich and List 2007a), which can be shown to generalize Arrow’s classic theorem. This result, in turn, builds on an earlier, seminal result on abstract aggregation by Nehring and Puppe (2002).. We restrict our attention to regular aggregation rules, defined as functions F : Un → U, which accept all profiles of complete and consistent individual judgment sets as admissible input (universal domain) and generate complete and consistent collective judgment sets as output (collective rationality)
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