Towards the definition of distance measures in the preference-approval structures

The task of combining preference rankings and approval voting is a relevant issue in social choice theory. The preference-approval voting (PAV) analyses the preferences of a group of individuals over a set of items. The main difference with the classical approaches for preference data consists in introducing, in addition to the ranking of candidates, a further distinction; candidates are subsetted in “acceptable” and “unacceptable”, or also in “good set” and “bad set” (a way to express the approval/disapproval). This work introduces the definition of a new measure to quantify disagreement between preference-approval profiles. For each pair of alternatives, we consider the two possible disagreements in approvals and rankings and merge them through a function h(·) increasing in each component. We show that our approach allows to emphasize particularly those cases in which both ranking and approval show simultaneously the maximum discordance.