Removal independent consensus methods for closed -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 : W n → 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 W n 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.