On some suggestions for having non-binary social choice functions
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The various paradoxes of social choice uncovered by Arrow , Sen  and others1 have led some writers to question the basic assumption of a binary social choice function underlying most of these paradoxes. Schwartz , for example, proves an important theorem which may be considered to be a generalization of the famous paradox of Arrow,2 and then lays the blame for this paradox on the assumption of a binary social choice function.3 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.4 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  paradox of a Paretian liberal, and thus a by-product of our analysis is a generalization of the theorem of Sen . The argument itself, however, is more general and applies also to the impossibility result proved by Schwartz .
See Murakami  for an exhaustive discussion of many of these paradoxes.
In a strictly formal sense, Schwartz's  theorem is not a generalization of Arrow's paradox in so far as Schwartz replaces some of Arrow's conditions by stronger conditions which cannot be deduced from any consistent subset of the set of Arrow conditions. The essence of Schwartz's theorem does, however, represent an extension of the paradox of Arrow. Also note that Schwartz  interprets his theorem not only in terms of collective decision-making but also in terms of individual decision-making. In this note we are concerned only with collective decision-making.
Another writer who discusses the condition of a binary choice function from a somewhat similar angle is A. Gibbard. In an unpublished paper he proves an important extension of Arrow's theorem and argues against the simultaneous insistence on a binary choice function and on Arrow's  condition of the independence of irrelevant alternatives. In this paper, however, we are not concerned with the condition of the independence of irrelevant alternatives.
It is, of course, being assumed that the best elements are defined for two-element sets.
KeywordsSocial Choice Choice Function Social Choice Function Impossibility Result Irrelevant Alternative
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