Consistent Bayesian Aggregation
The paper investigates the aggregation of first of all nonatomic subjective probabilities, second Savagean orderings, subject to the twofold consistency constraint that: (i) the aggregate is a subjective probability or a Savagean ordering, respectively; (ii) it satisfies the Pareto principle. Throughout the paper aggregation is viewed as a single profile exercise. In the case of nonatomic probabilities affine aggregative rules are the only solutions to the consistency problem; the coefficient sign may be determined by applying the stronger Pareto conditions (Propositions 1 and 2) . Speeial unanimity properties result from the assumption of nonatomicity (Proposition 3) . In the case of Savage an orderings even the existence of consistent solutions becomes a problem (Example 3). Under Pareto-indifference alone, as well as under any other Pareto condition when some minimum unanimity condition holds, solutions have to satisfy the overdetermined constraint that both the aggregate utility and the aggregate probability are affine in terms of the corresponding individual items (Propositions 4 and 6). This uniqueness result is shown to imply two Impossibility Theorems. Under Pareto indifference, as well as Weak Pareto when minimum unanimity prevails, affinely independent probabilities or utilities lead to some form of dictatorship (Proposition 5). Under Strong Pareto and minimum agreement the same independence assumptions lead to sheer impossibility unless either the utilities or the probabilities, respectively, are identical (Proposition 7). Nontrivial affine decompositions may exist in case of affine dependence (Example 4).
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