Succinctness of Languages for Judgment Aggregation

We review several different languages for collective decision making problems, in which agents express their judgments, opinions, or beliefs over elements of a logically structured domain. Several such languages have been proposed in the literature to compactly represent the questions on which the agents are asked to give their views. In particular, the framework of judgment aggregation allows agents to vote directly on complex, logically related formulas, whereas the setting of binary aggregation asks agents to vote on propositional variables, over which dependencies are expressed by means of an integrity constraint. We compare these two languages and some of their variants according to their relative succinctness and according to the computational complexity of aggregating several individual views expressed in such languages into a collective judgment. Our main finding is that the formula-based language of judgment aggregation is more succinct than the constraint-based language of binary aggregation. In many (but not all) practically relevant situations, this increase in succinctness does not entail an increase in complexity of the corresponding problem of computing the outcome of an aggregation rule.