Collective choice rules with social maximality

We consider social choice on a non-empty, finite set X where all elements of X are available; society will not be required to make a choice from a strict subset of X. Arrow assumes that the social preference relation is an ordering of the alternatives in X. However, under the assumption that the collective choice rule satisfies unrestricted domain, independence of irrelevant alternatives and the weak Pareto principle, this can be weakened to the requirement that (i) the social preference relation generates a non-empty set of maximal alternatives, and (ii) every maximal alternative is strictly socially preferred to every non-maximal alternative. Requirement (ii) can be weakened further when there are four or more alternatives. When only property (i) is satisfied, we characterize all collective choice rules that output complete social preferences and satisfy unrestricted domain, Pareto, neutrality and anonymity. These collective choice rules are related to the S rules of Bossert and Suzumura (2008). Finally, we prove some equivalence theorems relating the various binary relations considered in the paper to existing coherence conditions on social preferences.