Arrow's theorem, countably many agents, and more visible invisible dictators

Abstract

For infinite societies, Fishburn [Fishburn, P.C., 1970. Arrow's impossibility theorem: Concise proof and infinite voters. Journal of Economic Theory, 2: 103–106], Kirman and Sondermann [Kirman, A.P., Sondermann, D., 1972. Arrow's theorem, many agents, and invisible dictators. Journal of Economic Theory, 5: 267–277], and Armstrong [Armstrong, T.E., 1980. Arrow's theorem with restricted coalition algebras. Journal of Mathematical Economics, 7: 55–75] gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow's conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain `oracle' resolves Mihara's impossibility result (1997) about computability of social welfare functions.