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Suppose that a group of n individuals wishes to assign consensual probabilities to a sequence of propositions su..., sk which are pairwise contradictory (i^;^>(s, as, is logically false)) and exhaustive (si v... v sk is logically true). Suppose that after thorough discussion their opinions as to the most appropriate values of these probabilities are registered in an nxk matrix P=(p? 7), where ptj denotes the probability assigned by individual i to proposition s7. The terms of the problem dictate that the entries of each row of P be nonnegative and sum to one. Initial consensus on the values of the probabilities in question is reflected in a matrix with identical rows.

If consensus fails to obtain initially, there arises the problem of how to amalgamate the opinions in P into group estimates of the probabilities in question. One obvious possibility is to take a weighted arithmetic mean of the entries in each of the columns of P …