Linguistic majorities with difference in support

Majorities based on difference of votes and their extension, majorities based on difference in support, were introduced in social choice voting as tools to implement the crisp preference values (votes) and the intensities of preference provided by voters when comparing pairs of alternatives, respectively, with the aim to declare which alternative is socially preferred. Moreover, these rules require the winner alternative to reach a certain positive difference in its social valuation with respect to the one reached by the loser alternative. This paper introduces a new aggregation rule that extends majorities based on difference of votes from the context of crisp preferences to the framework of linguistic preferences. Under linguistic majorities with difference in support, the voters express their intensities of preferences between pairs of alternatives using linguistic labels and an alternative defeats another one if the first one reaches a specific support fixed before the election process. There exist two main representation methodologies of linguistic preferences: the cardinal one based on the use of fuzzy set, and the ordinal one based on the use of the 2-tuples. Linguistic majorities with difference in support are formalised in both representation settings, and conditions are given to guarantee that fuzzy linguistic majorities and 2-tuple linguistic majorities are mathematically isomorphic. Moreover, linguistic majorities with difference in support constitute a class of majority rules because all the possible majority rules can be generalised to the linguistic framework by adjusting the required threshold of support. Finally, linguistic majorities based on difference in support are proved to verify relevant normative properties: anonymity, neutrality, monotonicity, weak Pareto and cancellativeness.