On Irrelevance of Alternatives and Opinion Pooling

We consider the problem of combining k subjective probability distributions for a quantity of interest θ ∈ Θ. Two modified versions of the irrelevance of alternatives axiom are introduced by focusing on the consensual conditional odds, rather than the consensual probabilities. A strong modification of the axiom requires that the consensual odds depend only on the individual odds, while a weak modification requires that the consensual odds depend exclusively on the probabilities assigned by the experts to the two events being considered. We show that, provided some regularity conditions, the stronlgy modified axiom leads to characterization of the Logarithmic Opinion Pool. We also characterize the class of all the pools satisfying the weakly modified axiom. This class contains not only the Linear and the Logarithmic Pools but also pools that have been considered previously in the literature without axiomatic justifications.