The aggregation of imprecise probabilities

Two methods are presented for the aggregation of imprecise probabilities elicited from a group of experts in terms of betting rates. In the first method, the experts bet with a common opponent subject to limits on their personal betting stakes, and their individual and aggregate beliefs are represented by confidence-weighted lower and upper probabilities. This method is applicable to situations in which the events in question are discrete, few in number, and ambiguous in nature, where interest centers on the degree of consensus or dissensus and on the reconciliation of incoherence among a small group of experts. In the second method, the experts bet directly with each other as a means of reconciling incoherence, and their beliefs are represented by lower and upper risk neutral probabilities-i.e., products of probabilities and relative marginal utilities for money. The latter method is applicable to situations in which many individuals jointly assess probability distributions for large numbers of events or continuous random variables. In both methods, the fundamental concept of probability is an interpersonal one, with irreducible game-theoretic elements. (C) 2002 Elsevier Science B.V. All rights reserved.