Reformulating Arrow’s Conditions in Terms of Cardinal Pairwise Comparison Matrices Defined Over a General Framework

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 consistency and consistency. Then, we reformulate Arrow’s conditions in terms of PCMs, and we provide two preference aggregation procedures for representing group preferences that give a social PCM and a social cardinal ranking, respectively. Finally, we analyse how these preference aggregation procedures satisfy reformulated Arrow’s conditions.