Group Decis Negot (2018) 27:107–127 https://doi.org/10.1007/s10726-017-9552-8 Reformulating Arrow’s Conditions in Terms of Cardinal Pairwise Comparison Matrices Defined Over a General Framework Bice Cavallo 1 · Livia D’Apuzzo 1 · Gaetano Vitale 2 Published online: 14 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017 Abstract In the paper, we deal with cardinal preferences of experts when these are expressed by means of Pairwise Comparison Matrices (PCMs). In order to obtain general results, suitable for several kinds of PCMs proposed in literature, we focus on PCMs defined over a general unifying framework, that is an Abelian linearly ordered group. In this framework, firstly, we aggregate several PCMs and we analyse how the aggregated PCM preserves some coherence levels, such as transitivity, weak consis- tency and consistency. Then, we reformulate Arrow’s conditions in terms of PCMs, and we provide two preference aggregation procedures for representing group prefer- ences that give a social PCM and a social cardinal ranking, respectively. Finally, we analyse how these preference aggregation procedures satisfy reformulated Arrow’s conditions. Keywords Multi-criteria decision analysis · Pairwise comparison matrix · Con- sistency levels · Arrow’s conditions · Preference aggregation procedure · Abelian linearly ordered group B Bice Cavallo bice.cavallo@unina.it Livia D’Apuzzo liviadap@unina.it Gaetano Vitale gvitale@unisa.it 1 Department of Architecture, University of Naples “Federico II”, Naples, Italy 2 Department of Mathematics, University of Salerno, Fisciano, Italy 123