Removal independent consensus methods for closed β-systems of sets

Let β be a positive integer and let E be a finite nonempty set. A closed β-system of sets on E is a collection H of subsets of E such that A∈H implies |A|≥β, E∈H, and A∩B∈H whenever A,B∈H with |A∩B|≥β. If W is a class of closed β-systems of sets and n is a positive integer, then C:Wn→W is a consensus method. In this paper we study consensus methods that satisfy a structure preserving condition called removal independence. The basic idea behind removal independence is that if two input profiles P,P∗ in Wn agree when restricted to a subset A of E, then their consensus outputs C(P),C(P∗) agree when restricted to A. By working with the axiom of removal independence and classes of closed β-systems of sets we obtain a result for consensus methods that is in the same spirit as Arrow's Impossibility Theorem for social welfare functions.