Aggregating subjective probabilities: Some limitative theorems
Suppose that \(n\geq 2\) individuals assign subjective probabilities to a sequence of events \(E_ 1,...,E_ k\) which partition some set of possibilities. In recent years there has been extensive discussion of the sorts of restrictions which should be placed on methods for aggregating such probabilities into a single ''consensual'' distribution. In this paper we examine those aggregation methods which preserve independence, a restriction endorsed by \textit{R. Laddaga} [Synthese 36, 473-477 (1977; Zbl 0393.60003)] and others. (Preservation of independence requires that when all individuals assign probabilities to the atomic events \(E_ 1,...,E_ k\) so that some pair of events A and B in the algebra generated by \(\{E_ 1,...,E_ k\}\) turns out to be independent, A and B are independent with respect to the consensual distribution.) We prove that if the consensual probability assigned to each event \(E_ j\) is a function only of the probabilities assigned by individuals to that event, then the only independence preserving aggregation methods are either dictatorial (adopt the distribution of a single individual) or imposed (adopt an external distribution, irrespective of the distributions of any of the individuals). The above results have recently been generalized by Genest and the author to situations in which the consensual probability assigned to each atomic event \(E_ j\) is merely proportional to some function of the probabilities assigned by individuals to that event. In this context, when \(k=4\), a rich variety of non-dictatorial, non-imposed, independence preserving aggregation methods exist. When \(k\geq 5\), however, independence preservation again implies dictatorial or imposed aggregation.
independence preserving aggregation methods
scientific article; zbMATH DE number 3866358