their third best, etc. Now: some of the individual objects they vote for may
be retained, and others not. Likewise, some objects they do not vote for can
obtain. What matters for strategy proofness is whether the best set for each
agent among those that contain some externally fixed objects (those that are
chosen in spite of the agent’s negative vote) and do not contain some others
(those that are not chosen even if the agent supports them) is the set that
contains, in addition to those, as many elements from the agent’s preferred
feasible set. This is the case for additive preferences in all cases. It is also the
case for separable preferences if the first best for the agent is feasible, but not
necessarily otherwise. That is why, in the presence of infeasibilities, declaring
the best feasible set may not be a dominant strategy for some voters, even
when committees are used (except if the first best is always feasible, a situ-
ation studied in Barberà, Masso, and Neme (1997)). Whereas it is always a
dominant strategy for additive preferences. This accounts for the differences
in results under these two different domains.
We now make the definitions and the statements precise. Given a so-
cial choice function F: P" → 2κ and a subset B of Rf define the active
components of B in the range as
AC (B) = {A C B I X = Y ∩ BforsomeY ∈ Rf} .
Active components of B are subsets of B whose union with some subset in
Rf∖B is part of the range. Now, given B, Ç B Ç Rf define the range
complement of B' relative to B as
Cj (B') = {C C Rf∖B ∖B,UC E Rf} .
The range complement of a subset B' of B is the collection of sets in Rf∖B
whose union with B' is in the range. Notice that AC (B) can also be written
as {A C B I X U Y ∈ RpforsorneY ∈ Cf (X)}.
11
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