Aggregation of multiple prior opinions

Experts are asked to provide their advice in a situation of uncertainty. They adopt the decision maker’s utility function, but each has a potentially different set of prior probabilities, and so does the decision maker. The decision maker and the experts maximize the minimal expected utility with respect to their sets of priors. We show that a natural Pareto condition is equivalent to the existence of a set Λ of probability vectors over the experts, interpreted as possible allocations of weights to the experts, such that (i) the decision maker’s set of priors is precisely all the weighted-averages of priors, where an expert’s prior is taken from her set and the weight vector is taken from Λ; (ii) the decision maker’s valuation of an act is the minimal weighted valuation, over all weight vectors in Λ, of the experts’ valuations.