Continuous-Valued Binary Decision Procedures

Abstract

A Bergson-Samuelson social welfare function maps each n-tuple of continuous individual preference orderings into a continuous, transitive binary relation over alternative states of an economy. Arrow and his successors dropped the continuity requirement and demanded instead that the social ordering vary in a natural way with individual preferences. These requirements, the independence of irrelevant alternatives and citizen sovereignty axioms, together with the requirement that the social preference relation satisfy transitivity or some weaker rationality condition, imply either that power is concentrated to an extreme extent or that the social choice process is indecisive. In this paper we eliminate the transitivity assumption employed in the Bergson-Samuelson formulation and instead concentrate on the axiom that the social preference relation be continuous. Specifically, we examine the nature of a function that maps each n-tuple of individual preferences into a continuous binary relation that does not necessarily satisfy any rationality condition. When this function depends in a "natural" way on individual preferences, we show that power, although not necessarily concentrated, is distributed in a dichotomous fashion among individuals; each coalition can either determine the social preference relation on every pair of alternatives or on no pair of alternatives. Furthermore, any alternative x will be socially preferred to an alternative y only if those who prefer x to y constitute one of the powerful coalitions. Thus, such procedures have the form of a simple game.