Arrow-inconsistent economic domains

Abstract

In the public goods model of Kalai, Muller and Satterthwaite [8], any Arrow-consistent set of profiles of monotonic continuous preferences must be nowhere dense as a subset of the set of all such profiles in its usual topology. This result is the main theorem in [12], and suggests that the construction of SWFs in this economic environment should proceed from the relaxation of some Arrow axiom other than — i.e. in addition to — unrestricted domain. However, economic assumptions such as convexity induce nowhere dense sets of preferences, so if the social planner believes a priori in such restrictions, then the above prescription does not necessarily follow. This paper does for convexity, homotheticity, and (in the case of private goods) selfishness and even identity of tastes what [12] does for monotonicity and continuity. It shows that any arbitrarily small open N-fold product set of profiles satisfying any or all of these assumptions is an Arrow-inconsistent domain. Thus Arrow-consistent domains must be unreasonably small, so in constructing SWFs in these economic environments, one should be willing to relax some Arrow axiom other than unrestricted domain.