Voting by Committees under Constraints

This idea will play an important role in our characterization with additive
preferences. The next definition generalizes it to any pair of committees.

Definition 8 We say that two committees TV and TV^c are complementary
if D ∈ TV implies N∖D TV^c and D ∈ TV^c implies N∖D TV.

Example 2 Let K = {rr,y} be the set of objects and N = {1,2,3} the
set of agents. Assume that {0}, {ж}, and {y} are feasible but {rr,y} is not.
Consider the social choice function F defined by voting by quota 3 (which
respects feasibility) and let P be any additive (as well as separable) preference
profile such that τ(P₂) = τ(P₃) = {y} and {x, y}F₁{rr}F₁{y}F₁{0}. Since
^τ2^κ∖{x,y}(Pι) = {^}, У receives two votes and x one; therefore, F(P) = {0}.
However, if agent 1 declares the preference F₁' where {y}P[{x, y}P[{β}P[{x},
then у receives three votes and x none; that is, F(F₁',F₂,F₃) = {y}F_t{0} =
F(F₁,F₂,F₃). Hence, F is not strategy-proof.

The purpose of our two characterizations is to identify exactly the subfam-
ilies of committees that simultaneously respect feasibility and are strategy-
proof for the domains of additive and separable preferences.

We begin with some intuition about the nature of our results. For that, we
first remind the reader about the essential features of voting by committees
when there are no constraints, as in Barberà, Sonnenschein, and Zhou (1991).
There, the choice of a set can be decomposed into a family of binary choices,
one for each object. In each case, society decides whether the object should
or should not be retained, and the union of selected objects amounts to the
social alternative. If the methods used to decide upon each object are each
strategy-proof, then so is the method resulting from combining them into a
global decision, as long as the agent’s preferences are additive or separable.
Agents should be asked to express their best set, and under the expressed
domain restrictions this is equivalent to expressing those objects that they

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