COMBINING GROUP MEMBERS' JUDGMENTS
How should we combine the judgments of different people in order to obtain a representative judgment for the group? Such a group judgment must satisfy the reciprocal requirement: combining the judgments of all the people and then taking the reciprocal must give the same result as taking the reciprocal of each person’s judgment then combining them. We cannot use the arithmetic average because it does not satisfy this reciprocal requirement. For example, three people are estimating how many times an apple is larger than an orange. The first says it is twice larger, the second says three times larger, and the third says four times larger. The arithmetic average of their judgments is (2+3+4)/3 = 3, that is, the combined group judgment is 3 and its reciprocal is 1/3. To satisfy the reciprocity requirement it is necessary that the result also be 1/3 if we take the reciprocals of the individual judgments first then combine them using the same process of arithmetic averaging. But (1/2+1/3+1/4)/3 = 13/36 which is not 1/3. So the arithmetic average fails the reciprocity test and we cannot use it to combine group members’ judgments.
Since the arithmetic average does not work, we have one other simple alternative to try: the geometric average or geometric mean. To find the geometric average of n numbers multiply them together and take their nth root. The geometric average of the numbers 2, 3, and 4 is 2 x 3 x 4 = 24 raised to the power of 1/3. The reciprocal of this is 1/24 raised to the power of 1/3. Now we try it the other way around. Take reciprocals first, then multiply to obtain 1/2 x 1/3 x 1/4 = 1/24. Raising 1/24 to the power of 1/3 is the same number we obtained before by multiplying the numbers first then taking the reciprocal. So the geometric average satisfies the reciprocal requirement. Thomas Saaty has given a detailed mathematical proof that this is the proper way to combine group judgments (Thomas Saaty and Janos Aczel, "Procedures for Synthesizing Ratio Judgements," in the Journal of Mathematical Psychology, Vol. 27, No. 1, March 1983, pp. 93-102).
Individuals in a group, however, may possess different degrees of power as a result of many variables: formal authority within an organizational context, size and strength of an outside constituency, personal charisma, perceived intelligence, expertise on an issue, and the ability to call in favors owed by other group members. These factors, alone and in combination, make the judgments of some group members should count more than those of others. If the people themselves have priorities x, y, and z that sum to one indicating their respective importance as judges, then there is a method for combining the judgments analogous to the geometric mean that gives more weight to higher priority individuals. When three judges have equal power, each has a priority of 1/3 and the sum is one: 1/3 + 1/3 + 1/3 = 1. Taking the geometric mean of their judgments by multiplying them together then taking the cube root is the same as first raising each judgment to the power of 1/3 and then multiplying. So if the three judges instead have priorities of x, y, and z respectively with x + y + z = 1 we can, in an analogous way, raise each judgment to the power of the judge and multiply to get the combined judgment. For example, if their judgments are 2, 3 and 4, the combined judgment is 2 to the power of x times 3 to the power of y times 4 to the power of z.
To determine the priorities x, y, and z set up a hierarchy with criteria such as Wisdom, Experience, Previous Performance, Persuasive Abilities, and Effort on Problems with the judges as alternatives and perform pairwise comparisons in the usual way.
