Voting by Committees under Constraints

Proposition 2 Assume F: Sⁿ → 2^κ (or, F: Aⁿ → 2^κ) is strategy-proof.
Then, F is voting by committees.¹

3 Two Characterization Results

3.1 Examples and Preliminary Results

Because of feasibility constraints, not all combinations of committees can
be guaranteed to always generate a feasible outcome, even if all votes are
for feasible ones. The exact nature of the restrictions, i.e., the shape of
the range, will determine which combinations of committees can constitute
a proper social choice function for this range. Example 1 below illustrates
this fact. Moreover, under the presence of infeasibilities, there are voting
by committees that, although respecting feasibility, are not strategy-proof.
Example 2 illustrates this possibility.

Example 1 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.
Voting by quota 1 does not respect feasibility because for any preference
profile F, with the property that t(F_l) = τ(F₂) = {^} and τ(F3) = {y},
both x and у should be elected, which is infeasible. However, voting by
quota 2 does respect feasibility because x and у cannot get simultaneously
two votes (remember, agents cannot vote for infeasible outcomes) since the
complementary coalition of each winning coalition for x is not winning for y,
and viceversa.

¹It is easy to check that the proof of Proposition 2 in Barberà, Mass6, and Neme (1997)
which covers the case of separable preferences also aplies to the smaller domain of additive
preferences.

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