Transitive Majority Rule and the Theorem of the Alternative

Abstract

In this paper the theorem of the alternative is used to show the equivalence between transitive majority rule and the problem of finding positive solutions to a certain set of linear inequalities. A system of linear inequalities is derived whose possession of a strictly positive solution for every triple of alternatives is both necessary and sufficient for the transitivity of majority rule. The consequence of this is that we can consider not only restrictions on admissible individual preference patterns but also restrictions on the distribution of votes over these patterns to ensure transitivity.