1 Introduction
Many problems of social choice take the following form. There are n voters
and a set K = {1,..., k} of objects. These objects may be bills considered by
a legislature, candidates to some set of positions, or the collection of char-
acteristics 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, or if we are modelling the global results
of a legislature, which may pass or reject any number of bills. It is for these
cases that Barberà, Sonnenschein, and Zhou (1991) provided characteriza-
tions of all voting procedures which are strategy-proof and respect voter’s
sovereignty (all subsets of object may be chosen) when voters’ preferences
are additively represent able, 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. Rules in this class are defined by
a collection of families of winning coalitions, one for each object; agents vote
for sets of objects; to be elected, an object must get the vote of all members
of some coalition among those that are winning for that object.
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; if objects
are the characteristics of an alternative, some collections of characteristics
may be mutually incompatible, and others not. 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 represent able. Our main conclusions are the following. First: