Social welfare functions when preferences are convex, strictly monotonic, and continuous

Abstract

The paper shows that if the class of admissible preference orderings is restricted in a manner appropriate for economic and political models, then Arrow's impossibility theorem for social welfare functions continues to be valid. Specifically if the space of alternatives is R n, n > 3, where each dimension represents a different public good and if each person's preferences are restricted to be convex, continuous, and strictly monotonic, then no social welfare function exists that satisfies unanimity, independence of irrelevant alternatives, and nondictatorship. Arrow (1963) proved that for a set of at least three alternatives no nondictatorial social welfare function (SWF) exists satisfying unanimity (U) and independence of irrelevant alternatives (IIA), provided admissible preferences are not a priori restricted in some manner. If, however, the variety of preference orderings that are admissible is restricted sufficiently, then nondictatorial SWFs do exist that satisfy U and IIA. Single-peakedness, which Black (1948) discovered and Arrow (pp. 75-80) discussed, is the best known of these restrictions that is sufficient to make majority rule into a nondictatorial transitive SWF satisfying U and IIA. Papers of Inada (1969) and of Sen and Pattanaik (1969) generalized single-peakedness determined necessary and sufficient restrictions on the set of admissible preferences for majority rule to be a transitive SWF satisfying Arrow's conditions. Kramer (1973) used these results to show that majority rule is a valid Arrow type SWF only if the set of admissible preferences is restricted to a class that is much smaller than is justifiable by economic or political theory. These results describe the properties only of majority rule. The power of Arrow's theorem is that it rules out construction of any nondictatorial SWF satisfying U and IIA, not just social welfare functions based on majority rule. Our purpose in this paper is to show that the negative conclusions derived for the special case of majority rule generalize into true impossi