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. AGRIBUSINESS EXECUTIVE EDUCATION AND KNOWLEDGE EXCHANGE: NEW MECHANISMS OF KNOWLEDGE MANAGEMENT INVOLVING THE UNIVERSITY, PRIVATE FIRM STAKEHOLDERS AND PUBLIC SECTOR
2. HOW WILL PRODUCTION, MARKETING, AND CONSUMPTION BE COORDINATED? FROM A FARM ORGANIZATION VIEWPOINT
3. The name is absent
4. The name is absent
5. What Contribution Can Residential Field Courses Make to the Education of 11-14 Year-olds?
6. Multifunctionality of Agriculture: An Inquiry Into the Complementarity Between Landscape Preservation and Food Security
7. Can genetic algorithms explain experimental anomalies? An application to common property resources
8. The name is absent
9. Consciousness, cognition, and the hierarchy of context: extending the global neuronal workspace model
10. POWER LAW SIGNATURE IN INDONESIAN LEGISLATIVE ELECTION 1999-2004