Voting by Committees Under Constraints

Many problems of social choice take the following form. There are n voters and a set of k objects. These objects may be bills considered by a legislature, candidates to some set of positions, or the collection of characteristics which distinguish a social alternative from another. The voters must choose a subset of the set of objects. Sometimes, any combination of objects is feasible: for example, if we consider the election of candidates to join a club which is ready to admit as many of them as the voters choose. It is for these cases that Barbera, Sonnenschein, and Zhou (1991) provided characterizations of all voting procedures which are strategy-proof and respect voter's sovereignty when voters' preferences are additively representable, and also when these are separable. For both of these restricted domains, voting by committees turns out to be the family of all rules satisfying the above requirements. Most often, though, some combinations of objects are not feasible, while others are: if there are more candidates than positions to be filled, only sets of size less than or equal to the available number of slots are feasible. Our purpose in this paper is to characterize the families of strategy-proof voting procedures when not all possible subsets of objects are feasible, and voters' preferences are separable or additively representable. Our main conclusions are the following. First: all rules that satisfy strategy-proofness must still be voting by committees. Second: the committees for different objects must be interrelated, in precise ways which depend on what families of sets of objects are feasible. Third: unlike in Barbera, Sonnenschein, and Zhou (1991), the class of strategy- proof rules when preferences are additively representable can be substantially larger that the set of rules satisfying the same requirement when voter' preferences are separable.