A geometric examination of majorities based on difference in support

Reciprocal preferences have been introduced in the literature of social choice theory in order to deal with preference intensities. They allow individuals to show preference intensities in the unit interval among each pair of options. In this framework, majority based on difference in support can be used as a method of aggregation of individual preferences into a collective preference: option \(a\) is preferred to option \(b\) if the sum of the intensities for \(a\) exceeds the aggregated intensity of \(b\) by a threshold given by a real number located between 0 and the total number of voters. Based on a three dimensional geometric approach, we provide a geometric analysis of the non-transitivity of the collective preference relations obtained by majority rule based on difference in support. This aspect is studied by assuming that each individual reciprocal preference satisfies a \(g\)-stochastic transitivity property, which is stronger than the usual notion of transitivity.