Opinion pooling on general agendas
AbstractWe consider the classical problem of opinion pooling: the probability assignments of different individuals are to be merged into collective probability assignments. While opinion pooling has been explored in some depth in the literature (by statisticians, economists, and philosophers), all contributions so far assume that the set of relevant events (the "agenda") forms an algebra. However, the disjunction (union) or conjunction (intersection) of two relevant events need not be relevant in practice: it may well be relevant whether it rains, and relevant whether we are happy, but irrelevant whether the disjunction or conjunction of these two events holds. We explore opinion pooling without assuming that the agenda of relevant events forms an algebra. We show that, for a broad class of agendas (whose events may be much less interconnected than those of an algebra), any opinion pooling operator with two properties must be linear, i.e. derive the collective probability of each relevant event as a weighted linear average of the individuals'' probabilities of the event where the weights are event-independent. For an even broader class of agendas -- "nearly" all agendas -- we obtain a weaker conclusion: the pooling operator must be neutral, i.e. derive the collective probability of each relevant event as some (possibly non-linear) function of the individuals'' probabilities of the event where the function is event-independent.
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