This paper introduces a new axiom for choice in preference profiles and tournaments, called composition-consistency. A social choice function is composition-consistent if it is non-sensitive to the cloning of one or several outcomes. The key feature of the composition consistency property is an operation concept called multiple composition product of profiles. The paper provides a brief overview of some social choice functions studied in the literature. Concerning the tournament solutions, it is proved that the Top Cycle, the Slater and the Copeland solutions are not composition-consistent, whereas the Banks, Uncovered Set, TEQ, Minimal Covering Set are composition-consistent. Moreover, we define the compositionconsistent hull of a solution q~ as the smallest composition-consistent solution containing qS. The composition-consistent hulls of the Top cycle and Copeland solutions are specified, and we give some hints about the location of the hull of the Slater set. Concerning social choice functions, it is shown that Kemeny, Borda and Minimax social choice functions are not composition-consistent, whereas the Paretian one is composition-consistent. Moreover, we prove that the latter is the composition-consistent hull of the Borda and Minimax functions.
We consider situations of multiple referendum: finitely many yes-or-no issues have to be socially assessed from a set of approval ballots, where voters approve as many issues as they want. Each approval ballot is extended to a complete preorder over the set of outcomes by means of a preference extension. We characterize, under a mild richness condition, the largest domain of top-consistent and separable preference extensions for which issue-wise majority voting is Pareto efficient, i.e., always yields out a Pareto-optimal outcome. Top-consistency means that voters' ballots are their unique most preferred outcome. It appears that the size of this domain becomes negligible relative to the size of the full domain as the number of issues increases.
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