Voting by Committees under Constraints

Definition 5 A preference Pi on 2^κ is separable if for all XCK and all
y

X U {y} P_iXif and only if{y}Pβ.

Let S be the set of all separable preferences on 2^κ. We can give a geo-
metric interpretation to this set by identifying each object with a coordinate
and each set X of objects with a vertex of a ^-dimensional cube, i.e., with
the ^-dimensional vector of zeros and ones, where x belongs to X if and only
if that vector has a one in re’s coordinate. Sometimes we will make use of this
geometric interpretation. For instance, given X, Y C K the minimal box on
X and Y is the smallest subcube containing the vectors corresponding to X
and Y; namely,

MB (.V. У) = {Z ∈ 2^κ I (.V ∩ У) C Z C (.V U У)} .

Following with this interpretation, it is easy to see that a preference P_i is
separable if for all Z and Y ∈ MB (τ (P_i), Z) ∖ {Z}, YP₁Z.

Remark that additivity implies separability but the converse is false with
more than two objects. To see that, let K = {x,y,z} be the set of objects
and consider the separable preference

{x, У, ^z}P_i{y, z}P_i{x, z}P_i{x, y}P_i{x}P_i{y}P_i{z}P_i{Φ},
which is not additive since {^}Fj{y} and {y, z}P_i{x, z}. Geometrically, addi-
tivity imposes the condition that the orderings of all vertices on each parallel
face of the hypercube coincide while separability admits the possibility that
some vertices of two parallel faces have different orderings. This geometric
interpretation will become very useful to understand the differences of our
two characterizations.

To define voting by committees as in Barberà, Sonnenschein, and Zhou
(1991) we need the concept of a committee.

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