Remarks on the Theory of Collective Choice

Abstract

Ever since the so-called paradox of voting was generalized by Arrow (1963) to every democratic method of collective decision-making, a vast literature has appeared (a) trying to circumvent Arrow's difficulty by weakening some of his conditions (Bordes, 1976; Hansson, 1973; Plott, 1973; Sen, 1969); (b) proposing some other paradoxes in the theory of collective choice (Batra and Pattanaik, 1972; Hansson, 1969; Schwartz, 1970; Sen, 1970a) and (c) casting doubts about the relevance of Arrow's theorem to the theory of Paretian welfare economics (Bergson, 1966; Little, 1952; Samuelson, 1967, 1977). The purpose of this paper is to make some remarks on these recent developments in the theory of collective choice. The first part of the paper deals with the question of how much one needs to weaken Arrow's collective rationality condition in order to avoid his impossibility result. As is well known, Arrow (1963) imposed the collective rationality condition that the society can arrange all conceivable alternatives in order of preference and that, if some available set of alternatives is specified, the society must choose therefrom the best alternative with respect to that preference ordering. We will consider two conditions of consistent choice which are weaker than that of Arrow. The first condition requires that, if an alternative x is chosen over another alternative y in binary choice, y should never be chosen from any set of alternatives that contain x; while the second condition requires that, if x is chosen over y in binary choice, there exists no choice situation in which y is chosen and x is available but rejected. (In the second case y can be chosen if x is also chosen, while in the first case y cannot be chosen anyway.) There seems to be a gulf that separates possibility from impossibility in between these two seemingly similar consistency conditions. It will be shown that the first consistency requirement is incomnpatible with essentially Arrovian conditions on the collective choice rule, while the second consistency condition is compatible with the same conditions. Although the difference between these consistency requirements is very subtle, the implication thereof in the context of impossibility result is therefore dramatically different. Lest we should be too satisfied, we must hasten to add that no collective choice rule satisfying our second consistency requirement can be free from the paradox of Paretian liberal (Sen, 1970a; Batra and Pattanaik, 1972). The Arrow-Sen theory is then contrasted with the Bergson-Samuelson one. In view of doing this, it is convenient to remember that Arrow's incompatible conditions on collective choice rule can be classified into two categories. The first category consists of statements that apply to any fixed profile of individual preference relations, while the second category refers to the responsiveness of the collective choice to the variations in profiles. (The first category embraces the condition of collective rationality and the Pareto rule, while the second category consists of the independence of irrelevant alternatives