Path Independence, Rationality, and Social Choice

Abstract

The paper provides several axiomatizations of the concept of as applied to choice functions defined over finite sets. The axioms are discussed in terms of their relationship to postulates and their meaning with respect to social choice models. IN ANSWER to critics of the first edition of Social Choice and Individual Values, Arrow advanced in the second edition ajustification for imposing his consistency or which had not previously appeared, explicitly, in the social choice literature [2, p. 120]. He argued that the were necessary in order for social choices to be independent of the path of choice. He provided no real elaboration on the point. Perhaps he felt no explanation was necessary, since much of the social choice literature, especially those papers which deal with cycles, implicitly place a premium on some type of path-independence property. The purpose of this paper is to report some results which bear on the meaning and usefulness of this type of property. Specifically, it is shown that path independence is implied by, but does not imply rational choice. The importance of the observation is threefold. First, if path independence, rather than rationality, is desired as a property of social choice, the stronger need not be imposed. One result of this relaxation is that the immediate impossibility result discovered by Arrow is avoided. Welfare economists then are free to explore the possible applications of the tools he provided. Secondly, the observations made raise issues pertaining to the reasons for investigating mathematical properties like path independence in the first place. Thirdly, it is shown that the lines which separate properties, which induce immediate impossibility results, from path-independence properties are very thinly drawn. We will proceed as follows. Immediately below, in Section 2, a survey of the interpretations of the symbols is given. A glossary is also added at the end. Section 3 provides a brief summary of the arguments which have been advanced in support of rationality conditions in the case of social choice. These are presented in order that they can be separated from those arguments which are