In this paper we study to what extent majorities based on difference in support leads to triple-acyclic collective decisions. These majorities, which take into account voters' intensities of preference between pairs of alternatives through reciprocal preference relations, require to the winner alternative to exceed the support for the other alternative in a difference fixed before the election. Depending on that difference, i.e., on the threshold of support, and on some requirements on the individual rationality of the voters, we provide necessary and sufficient conditions for avoiding cycles of three alternatives on the collective decision.
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