Liberalism in the Theory of Social Choice

Abstract

There is an extensive literature concerning the conditions necessary for the existence of a social welfare function (see e.g. [1]) or a social choice function (see e.g. [4]) in the absence of 'any explicit principle of Liberalism; to this discussion we attempt no addition. However, Sen [3] has shown that the conditions become even more restrictive if such a principle is added in a particular form, and Gibbard [2], in discussing this and related problems, has uncovered yet more severe conditions. We are concerned in Sections 1 to 5 to analyse the way in which these difficulties arise and to argue that the way in which Liberalism is inserted into the problem has much to answer for. In Section 6 we propose a radically different way of incorporating Liberalism -an arguably more plausible waywhich does not lead to these difficulties. There is a set S of possible social states, each of which is a complete description of society, including every individual's place in it . Each individual i, i = 1, 2, ..., M, has a preference ordering Ri over S; an ordering is by definition complete, reflexive and transitive. The symbol Ri is used (without ambiguity) both to denote the ordering and in propositions like xRiy to denote pairwise preferences. The related symbols Pi and Ii are similarly used. A collective choice rule C is a function determining, for each M-tuple (R1, R2, ..., RM) of orderings on S a social preference relation R on S. (Again, R, P, and I are also used to denote pairwise preferences.) The properties of R are not specified, but for R to be either an ordering or a choice function it is necessary that it be acyclic. The authors quoted above show that, if Liberalism is introduced in a particular way, cycles of the form x1Px2, x2Px3, ..., xnPx1 can be generated, violating acyclicity. We define individual i as decisive between two social states x and y by the two implications