# Inconsistency and non-additive capacities: The Analytic Hierarchy Process in the framework of Choquet integration

Abstract

We examine the AHP in the framework of Choquet integration and we propose an extension of the standard AHP aggregation scheme on the basis of the Shapley values associated with the criteria. In our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the AHP. The measure of dominance inconsistency is then used to construct a non-additive capacity whose associated Shapley values reduce to the standard AHP priority weights in the consistency case. In the general inconsistency case, however, the extended aggregation scheme based on the Shapley weighted mean tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with respect to the other criteria.