Aggregating subjective probabilities: Some limitative theorems
type: Item
instance of: scholarly article
title: Aggregating subjective probabilities: Some limitative theorems
zbMATH Open document ID: 0544.62010
published in: Notre Dame Journal of Formal Logic
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62C99
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 60A05
zbMATH DE Number: 3866358
zbMATH Keywords: subjective probabilities
zbMATH Keywords: consensual probability
zbMATH Keywords: independence preserving aggregation methods
zbMATH Keywords: dictatorial
zbMATH Keywords: imposed
author: Carl G. Wagner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093870630
OpenAlex ID: W2066403266
DOI: 10.1305/NDJFL/1093870630
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description: scientific article; zbMATH DE number 3866358