The rarity of consistent aggregators

We demonstrate that the inconsistency associated with judgment aggregation, known as the “doctrinal paradox”, is not a rare exception. There are n individuals who have opinions about k propositions. Each opinion expresses the degree of belief or conviction and thus belongs to the unit interval [0,1]. We work with an arbitrary proposition aggregator that maps opinions about k propositions into an overall opinion in [0,1] and an arbitrary individual opinions aggregator mapping opinions of n individuals into a single judgement from a unit interval. We show that for any typical proposition aggregator, the set of individual opinion aggregators that are immune to the paradox is very small, i.e., is nowhere dense in the space of uniformly bounded functions. In addition, we offer several examples of judgement aggregation for which the paradox can be avoided. •Common assessments based on judges’ opinions might depend on the order of aggregation.•Opinions express degree of belief or conviction about propositions.•To get a common opinion one aggregates across propositions and then across individuals.•Alternatively, one might first aggregate across individuals.•If the results in the two cases are different, we have the “doctrinal paradox”.•Such a paradox is not a rare exception.•We show that the set of opinion aggregators immune to the paradox is meagre.•Linear aggregators and a few others are immune to the paradox.