General concepts of value restriction and preference majority

Regenwetter and Grofman offer a probabilistic generalization of Sen's classic value restriction condition when individual preferences are linear orders. They provide necessary and sufficient conditions for transitive majority preferences on linear orders. They call these conditions net value restriction and net preference majority. We study parallel generalizations for general binary relations. In general, neither net value restriction nor net preference majority is necessary for transitive majority preferences. Net value restriction is sufficient for transitive strict majority preferences, but not sufficient for transitive weak majority preferences. Net majority is sufficient for transitive majorities only if the preference relation with a net majority is a weak order. An application of our results to four U.S. National Election Study data sets reveals, in each case, transitive majorities despite a violation of Sen's original value restriction condition.