Beyond Electing and Ranking: Collective Dominating Chains, Dominating Subsets and Dichotomies

Classical voting rules output a winning alternative (or a nonempty set of tied alternatives). Social welfare functions output a ranking over alternatives. There are many practical situations where we have to output a different structure than a winner or a ranking: for instance, a ranked or non-ranked set of $k$ winning alternatives, or an ordered partition of alternatives. We define three classes of such aggregation functions, whose output can have any structure we want; we focus on aggregation functions that output dominating chains, dominating subsets, and dichotomies. We address the computation of our rules, and start studying their normative properties by focusing on a generalisation of Condorcet-consistency