CARLWAGNER
ALLOCATION, LEHRER MODELS, AND THE CONSENSUS OF PROBABILIq?IES 1. ALLOCATION PROBLEMS Rational choice often involves the assignment of values to numerical decision variables. In some cases the most appropriate values of these variables are obvious and one can move with dispatch to apply an appropriate optimization algorithm and identify an optimal choice or set of choices. In general, however, the determination of such values is a difficult task. Typically, individuals are required to make subjective estimates of probabilities and utilities or other predictions of future outcomes. The experience and sophistication required to quantify the relevant aspects of a decision problem thus often call for the expertise of more than one individual. But the strategy of decision-making by groups raises a further problem: What is to be done if the experts disagree?
Suppose that a group of n experts, labeled 1, 2 ..... n, is seeking numerical values of a sequence of k decision variables, x l, x~ ..... x k . The outcome of the group's deliberation is an n x k matrix A = (ate), where aij is the value assigned by expert i to variable xj. lntersubjective agreement on all of these values is reflected in a matrix with identical rows. Failing such agreement, and given the necessity of specifying a single value for each variable, how should the opinions registered in A be aggregated? Numerous possibilities from the realm of statistics (arithmetic and geometric means; medians; maxima, minima, and various combinations thereof) come to mind and have, indeed, frequently been employed in practice. Justifications for the choice of a particular method for aggregating group opinion have tended, however, to be piecemeal and anecdotal, relying heavily on tradition or simplicity of calcu!ation. 1 Our aim in this paper is to begin, in a modest way, to rectify this situation by presenting an axiomatic characterization of weighted arithmetic averaging as a method of combining group opinion for a special class of decision problems called allocation problems. Our main result (Theorem 4) may be specialized to provide both a formal foundation for Keith Lehrer's