The legitimacy of rank reversal (original) (raw)
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Rank Preservation and Reversal in Decision Making
Journal of Advances in Management Sciences & Information Systems, 2015
There are numerous real life examples done by many people which show that the alternatives of a decision sometimes can reverse their original rank order when new alternatives are added or old ones deleted and without bringing in new criteria. There is no mathematical theorem which proves that rank must always be preserved and there cannot be because of real life and hypothetical counter examples in decision making methods. Rank preservation came to be accepted as the standard because of techniques that could only rate alternatives one at a time treating them as independent. Thus an alternative receives a score and it will not change when other alternatives are added or deleted. All methods that only rate alternatives one at a time, thus always preserving rank, may not lead to the right decision; even if they may be right in certain areas of application. In reality, to determine how good an alternative is on an intangible criterion needs experience and knowledge about other alternatives and hence in their evaluation, the alternatives cannot be completely considered as independent of one another.
Rivista di Matematica per le Scienze Economiche e Sociali , 1997
The actual ranking of a set of alternatives is obtainable in a simple way assuming that the matrixA of pairwise comparisons isr-transilive. We show that, in some cases of inconsistency, the weights assigned to the alternatives by means of some well-known methods, suggested by the A.H.P., do not agree with the ranking. Further we introduce a condition, theweak consistency ofA, that ensures the mentioned methods provide weights according with the ranking.
Ranking and weak consistency in the A.H.P. context
Decisions in Economics and Finance, 1997
The actual ranking of a set of alternatives is obtainable in a simple way assuming that the matrix A of parwise comparisons is r-transitive. We show that, in some cases of inconsistency, the weights assigned to the alternatives by means of some well-known methods , suggested by the A.H.P, do not agree with the ranking. Further we introduce a condition, the weak concistency of A, that eusures the mentioned methods provide weights according with the ranking.
Pairwise ranking: choice of method can produce arbitrarily different rank order
2011
We examine three methods for ranking by pairwise comparison: Principal Eigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice of method can produce arbitrarily different rank order.To be precise, for any two of the three methods, and for any pair of rankings of at least four items, there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones. We discuss the implications of this result in practice, study the geometry of the methods, and state some open problems.
Generalized consistency and intensity vectors for comparison matrices
International Journal of Intelligent Systems, 2007
A crucial problem in a decision-making process is the determination of a scale of relative importance for a set X = {x1, x2,..., xn} of alternatives either with respect to a criterion C or an expert E. A widely used tool in Multicriteria Decision Making is the pairwise comparison matrix A = (aij), where aij is a positive number expressing how much the alternative xi is preferred to the alternative xj. Under a suitable hypothesis of no indifference and transitivity over the matrix A = (aij), the actual qualitative ranking on the set X is achievable. Then a vector w may represent the actual ranking at two different levels: as an ordinal evaluation vector, or as an intensity vector encoding information about the intensities of the preferences. In this article we focus on the properties of a pairwise comparison matrix A = (aij) linked to the existence of intensity vectors. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1287–1300, 2007.