The pairwise comparisons method is a convenient tool used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular implementations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and discrepancy of output. A global ranking discrepancy describes to what extent the obtained results correspond to the single expert's assessments. On the basis of the inconsistency and discrepancy indices, two properties of the weight deriving procedure are formulated. These properties are proven for Eigenvector Method and Koczkodaj's Inconsistency Index. Several estimates using Koczkodaj's Inconsistency Index for a principal eigenvalue, Saaty's inconsistency index and the Condition of Order Preservation are also provided.
IntroductionThe first documented uses of comparisons in pairs date back to the thirteenth century [4]. Later, the method was developed by Fechner [6], Thurstone [30] and Saaty [25]. The latter proposed the Analytic Hierarchy Process (AHP) extension to the pairwise comparisons (herein abbreviated as PC) theory, the framework allowing dealing with a large number of criteria.At the beginning of the twentieth century the method was used in psychometrics and psychophysics [30]. Now the method is considered part of decision theory [26]. Its utility has been confirmed in numerous examples [31,11,21,29]. Despite its long existence the area still prompts researchers to further exploration. Examples of such exploration are the Rough Set approach [10], voting systems [5], fuzzy PC relation handling [22,32], incomplete PC relation [2,7], nonnumerical rankings [14], nonreciprocal PC relation properties [9], rankings with the reference set of alternatives [18,19] and others. Further references can be found in [28,12].The aim of the PC method is to determine the weights of objects, so that the more important object is, the higher (or the lower) weight it has. The method provides the user with a number of specific techniques for deriving weights from the PC matrix M [3]. With every specific technique an appropriate inconsistency index is associated that describes to what extent M is inconsistent. The value of an inconsistency index is perceived as a kind of quality determinant for M -the input data to the deriving weight procedure. Following the popular adage "garbage in, garbage out" one could say that when the inconsistency is high (consistency is low) the result must be poor. Indeed, for various inconsistency indices there exist thresholds of acceptability, above which the obtained results are considered to be unreliable.However, poor and unreliable results may be also due to the deriving method itself. One way to learn something abou...