Decision-making with the AHP: Why is the principal eigenvector necessary

We will show here that the principal eigenvector of a matrix is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison consistent or near consistent matrix. A positive reciprocal n by n consistent matrix W = (wij) satisfies the relation wik = wij wjk . If the principal eigenvector of W is w=(w1 , ... ,wn ) the entries of W may be written as wij = wi /wj . Matrices A = (aij) that are obtained from W by small positive reciprocal perturbations of wij , known as near consistent matrices, are pivotal in representing pairwise comparison numerical judgments in decision making. Since W is always of the form wij = (wi/wj) a perturbation of W is given by aij = (wi/wj) εij and their corresponding reciprocals aji = (wj/wi) (1/εij). A priority vector for use in decision making that captures a linear order for the n elements compared in the judgment matrix can be derived for both consistent and near consistent matrices, but it is meaningless for strongly inconsistent matrice except if they are the result of random situations that have associated numbers such as game matches where the outcomes do not depend on judgment. It is shown that the ratios wi /wj of the principal eigenvector of the perturbed matrix are close to aij , if and only if the principal eigenvalue of A is close to n. We then show that if in practice we can change some of the judgments in a judgment matrix, it is possible to transform that matrix to a near consistent one from which one can then derive a priority vector. The main reason why near consistent matrices are essential is that human judgment is of necessity inconsistent, which if controllable, is a good thing. In addition, judgment is much more sensitive and responsive to large than to small perturbations, and hence once near consistency is reached, it becomes uncertain which coefficients should be perturbed with small perturbations to transform a near consistent matrix to a consistent one. If such perturbations were forced, they would seem arbitrary and can distort the validity of the derived priority vector in representing the underlying real world problem.