Voting by Committees under Constraints

Rewriting condition (??), we have

(У ∩ (B₁∖B₂)) U (A ∩ (B₂∖B₁)) U (A ∩ (B₁ ∩ B₂)) U (У ∩ B^c) ∈ R_f.

Therefore,
(A ∩ (B₂∖B₁)) U (A ∩ (B₁ ∩ B₂)) ∈ AC(B₂).           (4)

Then, by conditions (??) and (??), the fact again that B₂ is a section, and
Remark 2,

(A ∩ (B₂∖B₁)) U (A ∩ (B₁ ∩ B₂)) U (A ∩ (B₁∖B₂)) U (У ∩ B^c) ∈ R_f.

This implies that (A ∩ B) U (У ∩ B^c) ∈ R_f Hence, B is a section of R_f. ■
Proposition 3 Any set R_f has a unique minimal cylindric decomposition.
Proof Assume not. Let {B^,...,B^₁} and {B^,..., B^₂} be two distinct
minimal cylindric decompositions of R_f. There exists at least one pair
such that Bp_i ∩ B^₂ ≠ 0. By Lemma 2, B_j(_ι U B^₂ is a section of R_f. By
Lemma 1, Bp₁∖Bjζ is also a section of R_f implying, again by Lemma 1, that
{B^,..., B*₁} was not minimal.                                          ■

3.2 Additive Preferences

We can now state our first characterization.

Theorem 1 A social choice function F: Aⁿ → 2^κ is strategy-proof if and
only if it is voting by committees with the following properties:

(1) yV_x and Vff are equal for all x and y in the same active component of any
section with two active components in Rff s minimal cylindric decomposition,
(2) W_x and Vff are complementary for all x and y in different active compo-
nents of the same section in Rffs minimal cylindrical decomposition, when
there are only two active components in this section, and

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