an 'irrelevant' alternative should not and will not cause a change in the ranks of existing alternatives. The current version of the AHP can accommodate both these schools—its ideal mode preserves rank, while its distributive mode allows the ranks to change. Either mode is selected according to the problem at hand. Rank reversal and AHP are extensively discussed in a 2001 paper in Operations Research, as well as a chapter entitled Rank Preservation and Reversal, in the current basic book on AHP. The latter presents published examples of rank reversal due to adding copies and near copies of an alternative, due to intransitivity of decision rules, due to adding phantom and decoy alternatives, and due to the switching phenomenon in utility functions. It also discusses the Distributive and Ideal Modes of AHP. A new form of rank reversal of AHP was found in 2014 in which AHP produces rank order reversal when eliminating irrelevant data, this is data that do not differentiate alternatives. There are different types of rank reversals. Also, other methods besides the AHP may exhibit such rank reversals. More discussion on rank reversals with the AHP and other MCDM methods is provided in the rank reversals in decision-making page. Non-monotonicity of some weight extraction methods Within a comparison matrix one may replace a judgement with a less favorable judgment and then check to see if the indication of the new priority becomes less favorable than the original priority. In the context of tournament matrices, it has been proven by Oskar Perron that the principal right eigenvector method is not monotonic. This behaviour can also be demonstrated for reciprocal n x n matrices, where n > 3. Alternative approaches are discussed elsewhere. See also Analytic hierarchy process – car example Analytic hierarchy process – leader example Analytic network process Arrow's impossibility theorem Decision making Decision-making paradox Decision-making software Hierarchical decision process L. L. Thurstone Law of comparative judgment Multi-criteria decision analysis Pairwise comparison Preference Principal component analysis Rank reversals in decision-making References Further reading Saaty, Thomas L. Decision Making for Leaders: The Analytical Hierarchy Process for Decisions in a Complex World (1982). Belmont, California: Wadsworth. ; Paperback, Pittsburgh: RWS. . "Focuses on practical application of the AHP; briefly covers theory." Saaty, Thomas L. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process (1994). Pittsburgh: RWS. . "A thorough exposition of the theoretical aspects of AHP."