The existence of group preference functions

Abstract

Although Kenneth Arrow's impossibility theorem in social choice theory ([1]) is well-known and famous, it seems to be less common knowledge that this result does not hold for the case of an infinite number of voters. As far as I know, the first time this fact has been mentioned in print is in Peter Fishburn's [3]. There he gave a proof of the existence of what Arrow called a "social welfare function", using a special kind of probability measure. In a letter Peter Fishburn has informed me that Julian Blau knew about the infinite voter result in 1960 already. He has, however, not published anything on it. In this paper I will prove that not only do such functions exist, but they exist in great numbers. In fact, the cardinal number of functions satisfying the conditions posed by Arrow equals the total number of functions from the same domain. Arrow's theorem will follow as a corollary when the main theorem is combined with a simple mathematical fact. The roles played by the different conditions are spelt out more clearly than usual when Arrow's result is proved in this way. The technique of the proof is also applicable to the case of so-called "quasi-transitive" social preference. Especially Amartya Sen [8] and Frederic Schick [7] have argued that a possible solution to the problem of social choice would be to relax the condition that social preference be transitive and require only