Notes on discrepancy in the pairwise comparisons method
This work addresses a theoretical issue in decision-making methods for researchers and practitioners, but it is incremental as it builds on prior critiques.
The paper tackles the problem of order preservation in the pairwise comparisons method, showing that even consistent matrices may not meet the condition of order preservation (COP), and it defines a more precise criteria using a discrepancy factor to determine when COP is met.
The pairwise comparisons method is a convenient tool used when the relative order among different concepts (alternatives) needs to be determined. One popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix. In such cases the ranking result the principal eigenvector of the pairwise comparison matrix is adopted, whilst the eigenvalue is used to determine the index of inconsistency. A lot of research has been devoted to the critical analysis of the eigenvalue based approach. One of them is the work (Bana e Costa and Vansnick, 2008). In their work authors define the conditions of order preservation (COP) and show that even for a sufficiently consistent pairwise comparisons matrices, this condition can not be met. The present work defines a more precise criteria for determining when the COP is met. To formulate the criteria a discrepancy factor is used describing how far the input to the ranking procedure is from the ranking result.