Voting by Committees under Constraints



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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