Voting by Committees under Constraints

Q

f∖...,f^b< are strategy-proof and F (F) = ∣J F^βp (P^βp} for all P ∈Aⁿ.
p=l

(b) Conversely, let {Bι,..., B_q} be a partition of K' Ç K and let {Bι, ...,B_q}
be a collection of subsets of objects, with B_p Ç 2^βp for all p = 1, ...,g. Let
F^βp: Ap —> B_p be a collection of onto social choice functions, one for each
p = l,...,q. If F^βl,...,F^βq are strategy-proof, then the function F (F) =
g

∣J F^βp (P^βp} for all P ∈A" is strategy-proof, {B_i,..., B_q} is a cylindric
p=l

decomposition of Rf = K', and Rf = B_i + ... + B_q.

Proof (a) Assume {B_i,..., B_q} is a cylindric decomposition of Rf and
let F ∈A". Then,

by definition of Rf

since {B₁,..., B_q} is a partition of R_f

by definition of F^βp and P^βp.

To obtain a contradiction, assume that F^βp is not strategy-proof; that is,
there exist P^βp, i, and P^Bp such that F^Bp(P^Bp, P^Bp')P^BpF^βp(P^βp'). There-
fore, and since preferences are additive,

∑ ^u*^p(^y) > Σ ^uf4y)> (ð)
yeF^Bp(pf^p ,pf^p)             yeF^Bp(P^Bp)

for any uf^p : B_p → IR representing P^Bp.

Take any F ∈ Aⁿ generating P^βp and P_i generating P^Bp with the property
that

F' = F'                   (6)

for all p^, ≠ p. For each p^, ≠ p, take any uf^p' representing P^Bp'. Then, by
condition (??),

£   £  u⅛(₁) +  £  √'⅛)

P^z≠P x(PF^bp' (P^bp' )               yeF^BP(P^BP)

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