Remarks on the Theory of Collective Choice.

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 incompatible with essentially Arrovian conditions... [ABSTRACT FROM AUTHOR]