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Consistent Bayesian Aggregation

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  • Mongin Philippe

Abstract

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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Suggested Citation

  • Mongin Philippe, 1995. "Consistent Bayesian Aggregation," Journal of Economic Theory, Elsevier, vol. 66(2), pages 313-351, August.
  • Handle: RePEc:eee:jetheo:v:66:y:1995:i:2:p:313-351
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    1. Sugden Robert, 1993. "An Axiomatic Foundation for Regret Theory," Journal of Economic Theory, Elsevier, vol. 60(1), pages 159-180, June.
    2. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
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    7. Machina, Mark J, 1987. "Choice under Uncertainty: Problems Solved and Unsolved," Journal of Economic Perspectives, American Economic Association, vol. 1(1), pages 121-154, Summer.
    8. Heath, Chip & Tversky, Amos, 1991. "Preference and Belief: Ambiguity and Competence in Choice under Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 4(1), pages 5-28, January.
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    10. Munier, Bertrand, 1989. "New models of decision under uncertainty: An interpretative essay," European Journal of Operational Research, Elsevier, vol. 38(3), pages 307-317, February.
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