Ranking by eigenvector versus other methods in the analytic hierarchy process

Counter-examples are given to show that in decision making, different methods of deriving priority vectors may be close for every single pairwise comparison matrix, yet they can lead to different overall rankings. When the judgments are inconsistent, their transitivity affects the final outcome, and must be taken into consideration in the derived vector. It is known that the principal eigenvector captures transitivity uniquely and is the only way to obtain the correct ranking on a ratio scale of the alternatives of a decision. Because of this and of the counter-examples given below, one should only use the eigenvector for ranking in making a decision.