Q
f∖...,fb< are strategy-proof and F (F) = ∣J Fβp (Pβp} for all P ∈An.
p=l
(b) Conversely, let {Bι,..., Bq} be a partition of K' Ç K and let {Bι, ...,Bq}
be a collection of subsets of objects, with Bp Ç 2βp for all p = 1, ...,g. Let
Fβp: Ap —> Bp be a collection of onto social choice functions, one for each
p = l,...,q. If Fβl,...,Fβq are strategy-proof, then the function F (F) =
g
∣J Fβp (Pβp} for all P ∈A" is strategy-proof, {Bi,..., Bq} is a cylindric
p=l
decomposition of Rf = K', and Rf = Bi + ... + Bq.
Proof (a) Assume {Bi,..., Bq} is a cylindric decomposition of Rf and
let F ∈A". Then,
F(P)
= F(P) ∩ Rf
Q
= ∪ [F(P) ∩ Br]
p=l
g
= U Fβp{Pβp')
p=l
by definition of Rf
since {B1,..., Bq} is a partition of Rf
by definition of Fβp and Pβp.
To obtain a contradiction, assume that Fβp is not strategy-proof; that is,
there exist Pβp, i, and PBp such that FBp(PBp, PBp')PBpFβp(Pβp'). There-
fore, and since preferences are additive,
∑ u*p(y) > Σ uf4y)> (ð)
yeFBp(pfp ,pfp) yeFBp(PBp)
for any ufp : Bp → IR representing PBp.
Take any F ∈ An generating Pβp and Pi generating PBp with the property
that
F' = F' (6)
for all p, ≠ p. For each p, ≠ p, take any ufp' representing PBp'. Then, by
condition (??),
£ £ u⅛(1) + £ √'⅛)
Pz≠P x(PFbp' (Pbp' ) yeFBP(PBP)
22
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