the other with top on 0 (orderings {ж} >~4 B >-4 D, B >b D >-δ {ж}, and
B {x} D).
Proof of Theorem 1 To prove necessity, let F∙. An —> 2κ be a strategy-
proof social choice function and let {Bi,..., Bq} be the minimal Cylindric
decomposition of ⅛, which exists by Proposition 3.
(1) Assume that x,y ∈ Zi ∈ AC(Bp) = {Z1,Z2}. Since {Bi,..., Bq} is
minimal we have that Zi ∩ Z2 = 0. Assume that W''' ≠ TV™; that is, there
exists I ∈ W''' such that I TV™. Consider any P such that τ(Pi)Γ∖Bp = Z1
for all i ∈ I and τ(Pf) ∩ Bp = Z2 for all j ∈ N∖I. Then, x ∈ F(P) and
у <f F(P) contradicting that x and у belong to the same active component
of Bp.
(2) Assume x ∈ A, у ∈ Y, and AC(Bp) = {A, У}. To obtain a contra-
diction assume there exists D ∈ W''' and N∖D ∈ TV™. It is easy to find P
such that x, у ∈ F(P) contradicting that x and у belong to different active
components of Bp.
(3) Follows from part (a) of Proposition 4 and Proposition 5.
Sufficiency follows from part (b) of Proposition 4, since it is clear that
all social choice functions defined on each of the sections are onto the active
components of the section and strategy-proof.
6 References
ANSWAL, N., CHATTERJI, S., and SEN, A. (1999). “Dictatorial Do-
mains”, Discussion paper 99-05, Indian Statistical Institute.
BARBERÀ, S., GUL, F., and STACCHETTI, E. (1993). “Generalized Me-
dian Voter Schemes and Committees”, Journal of Economic Theory, 61,
262-289.
28
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