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Removal independent consensus methods for closed \(\beta \)-systems of sets. (English) Zbl 1176.91029

Summary: Let \(\beta \) be a positive integer and let \(E\) be a finite nonempty set. A closed \(\beta \)-system of sets on \(E\) is a collection \(H\) of subsets of E such that \(A\in H\) implies \(|A|\geq \beta , E\in H\), and \(A\cap B\in H\) whenever \(A,B\in H\) with \(|A\cap B|\geq \beta \). If \(\mathcal W\) is a class of closed \(\beta \)-systems of sets and \(n\) is a positive integer, then \(C : \mathcal W^n \rightarrow W\) 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 \(\mathcal W^n\) 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 \(\beta \)-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.

MSC:

91B12 Voting theory
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