On some suggestions for having non-binary social choice functions

The various paradoxes of social choice uncovered by Arrow 1], Sen 10] and others have led some writers to question the basic assumption of a binary social choice function underlying most of these paradoxes. Schwartz 8], for example, proves an important theorem which may be considered to be a generalization of the famous paradox of Arrow, and then lays the blame for this paradox on the assumption of a binary social choice function. He then proceeds to define a type of choice functions which, like binary choice functions, define the best elements in sets of more than two alternatives on the basis of binary comparisons, but which, as he claims, have an advantage over binary choice functions, in so far as they always ensure the existence of best elements for sets of more than two alternatives irrespective of the results of binary comparisons. The purpose of this paper is to show that even a considerable weakening of the assumption of a binary social choice function does not go very far towards solving some of the paradoxes under consideration, and that if replacing the requirement of a binary social choice function by a Schwartz type social choice function solves these paradoxes, it does so only by violating the universally acceptable value judgment that in choosing from a set of alternatives, society should never choose an alternative which is Pareto inoptimal in that set (i.e., the socially best alternatives in a set should always be Pareto optimal). This argument is substantiated with the help of an extended version of Sen's 10] paradox of a Paretian liberal, and thus a by-product of our analysis is a generalization of the theorem of Sen 10]. The argument itself, however, is more general and applies also to the impossibility result proved by Schwartz 8].We are extremely grateful to Amartya Sen for his helpful comments.

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