would prefer to be included in the social decision, rather than not.
In our case, a first difference is that the choice of sets may not be decom-
posable to the extreme of allowing for independent decisions on each object.
Our results tell us precisely about the extent to which global decisions can
be decomposed, and say how to coordinate the decisions within groups of
objects that require joint treatment. Indeed, in the presence of infeasibili-
ties, the decision on what objects to choose, and which ones not to, can no
longer be decomposed into object-by-object binary decisions. For example,
choosing x might only be possible if y is not chosen: then the choices regard-
ing x and y must be joint. Similarly, z might only be chosen if w is, and
again decisions involving these two objects need to be coordinated. Yet, if all
feasible choices of x and y, when coupled with any feasible choice for z and
w, turn out to be feasible, there is still room for decomposition of the choices
in two blocks of objects. If, on the contrary, further restrictions must take
into account, whereby certain feasible choices from x and y become incom-
patible with some feasible choices from z and w, then decomposition is not
possible. The paper provides a precise statement about the extent to which
decisions on what sets to choose can be decomposed into partial decisions
involving subsets (we call each part of the decomposition a section), in the
presence of feasibility constraints. Moreover, we discuss the characteristics
of the committees that must be used in order to coordinate the choices of
objects within each of the sections.
A second contrast with the unconstrained case is that our results for
separable preferences are quite different (and much more negative) than for
additive preferences. Essentially, this is because in the presence of infeasibil-
ities, agents are not asked to vote for their preferred sets, but rather for their
preferred feasible sets. Hence, they may end up voting for their second best,
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