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

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

Fb (Pb) = F(P)∏ B,

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

Remark 4 Notice that, since F. An —> 2κ 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 Fb = Fb.
Therefore, Fb is well-defined.

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

21



More intriguing information

1. Needing to be ‘in the know’: strategies of subordination used by 10-11 year old school boys
2. 09-01 "Resources, Rules and International Political Economy: The Politics of Development in the WTO"
3. Experimental Evidence of Risk Aversion in Consumer Markets: The Case of Beef Tenderness
4. The name is absent
5. MANAGEMENT PRACTICES ON VIRGINIA DAIRY FARMS
6. The name is absent
7. Population ageing, taxation, pensions and health costs, CHERE Working Paper 2007/10
8. The name is absent
9. The name is absent
10. Evaluating Consumer Usage of Nutritional Labeling: The Influence of Socio-Economic Characteristics