A logical interpretation for the eigenvalue method in the analytic hierarchy process

The eigenvalue method (EM), that is, to find the principal eigenvector of a pairwise comparison matrix, is widely used and known to be practical in Analytic Hierarchy Process (AHP). However, the validity of EM has never been fully proved. In this article, we have justification for using EM in AHP. By introducing two concepts, self-evaluation and non-self-evaluation, into AHP, the fundamental theorem (Frobenius' Theorem) for EM is interpreted as two optimization problems. From these two concepts, a noncooperative game with a pairwise comparison matrix is also formulated and its equilibrium solution is the principal eigenvector. We propose two discrepancy indices between self-evaluation and non-self-evaluation and formulate four discrepancy-minimization problems. An optimal solution for two minimization problems among them is equal to the principal eigenvector.