A Note on Arrow's Postulates for a Social Welfare Function--a Comment

Abstract

In a note on Arrow's (1963, chap. iii) five individually reasonable but collectively incompatible postulates for a social welfare function, Bruno Contini (1966) had sought to prove the following theorem: The condition of positive association of social and individual values is always vacuously satisfied if the choice space is a connected space. The implication of this theorem, as Contini hopefully pointed out, is that this postulate of Arrow's is irrelevant and can be dispensed with when the choice space is connected. It is the purpose of this comment to show (a) that Contini's theorem as stated above is false; (b) that while the theorem that Contini succeeded in proving has to assume the conditions for the existence of a continuous real-valued utility function defined over the choice space, a similar theorem, using more acceptable conditions., which will also cover some cases of even lexicographically ordered choice spaces can be stated; and, finally (c) that similar theorems, far from encouraging us to dismiss the positive-association postulate should, in fact, help identify the important types of choice spaces for which the Arrow postulate will be non-trivial, and hence really relevant.