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 H7m the set of its minimal
winning coalitions. A committee H7 is dictatorial if there exists i ∈ N such
that H7m = {{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 —> 2κ is voting by com-
mittees, if for each x ∈ K, there exists a committee Wx such that for all
P = (P- P } tfr'
x ∈ F(p~)if and only if{i ∈ N ∣ x ∈ т-лР[Р()} ∈ Wx.
A social choice function F is called Voting by quota q (1 ≤ q < n) if for
all x the committee Wx 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 → 2κ (or, F: An → 2κ)
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.