Approximately classic judgement aggregation

This paper analyzes judgement aggregation problems in which a group of agents independently votes on a set of complex propositions subject to an interdependency constraint. It considers the issue of judgement aggregation from the perspective of approximation; that is, it generalizes the classic framework of judgement aggregation by relaxing the two main constraints assumed in the literature, Consistency and Independence. In doing so, it also considers mechanisms that only approximately satisfy these constraints, that is, satisfy them up to a small fraction of the inputs. The main question raised is whether the relaxation of these constraints significantly alters the class of aggregation mechanisms that meet the two (relaxed) constraints. The main result of this paper is that in the case of a subclass of a natural class of aggregation problems termed “truth-functional agendas,” the set of aggregation mechanisms that meet the constraints does not extend nontrivially when the constraints are relaxed. This paper also shows connections between this new general framework and the works on approximation of preference aggregation as well as the field of Property Testing and particularly linear testing of Boolean functions.