Probabilistic opinion pooling generalized. Part one: general agendas

How can di¤erent individuals’probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events –the agenda –is a -algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain or an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are -algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we unify our results with existing results on binary judgment aggregation and Arrovian preference aggregation. Our uni…ed theorems show why the same kinds of axioms (independence and consensus preservation) have radically di¤erent implications for di¤erent aggregation problems: linearity for probability aggregation and dictatorship for binary judgment or preference aggregation. Keywords: Probabilistic opinion pooling, judgment aggregation, subjective probability, probabilistic preferences, vague/fuzzy preferences, agenda characterizations, a uni…ed perspective on aggregation Although both authors are jointly responsible for this paper and project, Christian List wishes to note that Franz Dietrich should be considered the lead author, to whom the credit for the present mathematical proofs is due. This paper is the …rst of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. Both papers build on our earlier, unpublished paper ‘Opinion pooling on general agendas’(September 2007).