Voting by Committees under Constraints

Definition 6 A committee H⁷ is a nonempty family of nonempty coalitions
of N, which satisfies coalition monotonicity: if I ∈ W and IQJ, then
J ∈ W. Coalitions in H⁷ are called winning. A coalition I ∈ H⁷ is a
minimal winning coalition if for all J Q I we have that J W.

Given a committee H⁷, we will denote by H^7m the set of its minimal
winning coalitions. A committee H⁷ is dictatorial if there exists i ∈ N such
that H^7m = {{i}}. Associated to each family of committees (one for each
object) we can define a special type of social choice functions.

Definition 7 A social choice function F : Pⁿ —> 2^κ is voting by com-
mittees, if for each x ∈ K, there exists a committee W_x such that for all
P = (P- P } tf^r'

x ∈ F(p~)if and only if{i ∈ N ∣ x ∈ т-л_Р[Р()} ∈ W_x.

A social choice function F is called Voting by quota q (1 ≤ q < n) if for
all x the committee W_x is equal to the family of coalitions with cardinality
equal or larger than q.

We state, as Proposition 1 below, Barberà, Sonnenschein, and Zhou
(1991) ^,s characterization of voting by committees as the class of strategy-
proof social choice functions on S, as well as on A, satisfying voter’s sovereignty.

Proposition 1 A social choice function F: Sⁿ → 2^κ (or, F: Aⁿ → 2^κ)
is strategy-proof and satisfies voter’s sovereignty if and only if it is voting by
committees.

To cover social choice problems with constraints we have to drop the
voter’s sovereignty condition of Proposition 1. But a result in Barberà,
Masso, and Neme (1997) tells us that the only strategy-proof rules in this
case must still be of the same form: this is stated in Proposition 2.

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