Strategy-proof aggregation of approximate and imprecise judgments

We study strategy-proofaggregation rules for exact and approximate judgments of agents when the judgement space is a bounded distributivelattice. We start from the observation that judgment spaces having such a structure are endowed with an intrinsic metric which makes it possible and natural to impute single-peaked preferences to agents. Moreover, we show that bounded distributivity of the judgment space guarantees that the same structure is inherited by its order intervals. Relying on such facts we prove the existence of a large class of inclusive and unanimity-respecting strategy-proof aggregation rules for both exact and approximate judgments. Amongst them, the simple majority aggregation rule is characterized as the only one that satisfies both anonymity and bi-idempotence (i.e. ensures a definite choice between the two judgments nominated by a maximally polarized body). Finally, we consider several applications of our results including approximate probability estimates as modeled by intervals of probability values, numerical measurements with explicit error bounds, approximate classifications, and conditional judgments that are amenable to composition by means of a set of logical connectives.