Voting by Committees under Constraints

considering the universal domain assumption of the Gibbard-Satterthwaite
result. Theorem 1 tells us that only dictatorial rules are strategy-proof on
additive preferences. This is the conclusion we wanted.

5 Appendix: Proof of Theorem 1

The proof of Theorem 1 is based on a decomposition argument that applies an
important result of Le Breton and Sen (1997) to our context. This argument,
which will be exploited in the proof of Theorem 1, is expressed as Proposition
4 below. But before, we need the following notation.

Let Pi be an additively representable preference on 2^κ and consider a
subset B of K. Let Pf stand for the preferences on 2^b generated by the
utilities which represent Pi. Let Aβ be the set of additive preferences on 2^b.
For a profile P of preferences on 2^κ, P^b will denote the profile of preferences
so restricted, for all i ∈ N.

Given a strategy-proof social choice function F: Aⁿ → 2^κ and a subset
B of objects, let F^b: A⅞ —> 2^b be defined so that for all P^b ∈A⅛

F^b (P^b) = F(P)∏ B,

where P is any additive preference such that P^b is generated by the utilities
which represent P.

Remark 4 Notice that, since F. Aⁿ —> 2^κ is a strategy-proof social choice
function, it is voting by committees (by Proposition 2). Hence, for any
B C K, F(P) ∩ B = F (p} ∩ B for all F, F ∈A" such that F^b = F^b.
Therefore, F^b is well-defined.

Proposition 4 (a) Let F: Aⁿ 2^κ be a social choice function and let
{Bι,..., B_q} be a cylindric decomposition of Rp. If F is strategy-proof then

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