Consistent Bayesian Aggregation

We investigate the aggregation of first nonatomic probabilities and second Savagean orderings, subject to the following consistency constraints: (i) the aggregate is a subjective probability or a Savagean ordering, respectively; and (ii) it satisfies the Pareto principle. Aggregation is viewed here as a single-profile exercise. We show that affine rules are the only solutions to the problem of aggregating nonatomic probabilities. Assuming weak Pareto conditions and technical restrictions, we show that dictatorial rules are the only solutions to the problem of aggregating Savagean orderings. No solution exists when strong Pareto replaces weaker conditions. Without technical restrictions of affine independence, nontrivial solutions may exist. (C) 1995 Academic Press, Inc.