On the extraction of weights from pairwise comparison matrices

We study properties of weight extraction methods for pairwise comparison matrices that minimize suitable measures of inconsistency, ‘average error gravity’ measures, including one that leads to the geometric row means. The measures share essential global properties with the AHP inconsistency measure. By embedding the geometric mean in a larger class of methods we shed light on the choice between it and its traditional AHP competitor, the principal right eigenvector. We also suggest how to assess the extent of inconsistency by developing an alternative to the Random Consistency Index, which is not based on random comparison matrices, but based on judgemental error distributions. We define and discuss natural invariance requirements and show that the minimizers of average error gravity generally satisfy them, except a requirement regarding the order in which matrices and weights are synthesized. Only the geometric row mean satisfies this requirement also. For weight extraction we recommend the geometric mean.