Sets of Alternatives as Condorcet Winners

Choice of more than one alternative from a set of available alternatives, a committee for instance, is often carried out by applying certain choice rules to preference profiles where preferences are defined over alternatives. However, since preferences towards sets of alternatives, are not considered, social qualities of the final outcome remain problematic. Direct verification of such qualities is impractical since, generally, information on preferences on sets of alternatives is only partially available or not available at all. This paper attempts to solve this problem by connecting preferences over alternatives to preferences towards sets of such alternatives via certain extension axioms. We concentrate on two sorts of committees, one that will necessarily be a Condorcet winner at all admissible preference extensions, and one that will be a Condorcet winner at some admissible extension. Former requirement turns out to be too demanding, in fact impossible as is, even when the set of admissible preference extensions is fairly narrow. Nonetheless, when the choice problem is restricted to sets of same cardinality, a full characterization is proven possible. Second requirement enables more permissive results and we characterize these sets by using different versions of a non-domination condition