Voting by Committees under Constraints



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

X U {y} PiXif and only if{y}Pβ.

Let S be the set of all separable preferences on 2κ. 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 2κ I (.V ∩ У) C Z C (.V U У)} .

Following with this interpretation, it is easy to see that a preference Pi is
separable if for all
Z and Y MB (τ (Pi), 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}Pi{y, z}Pi{x, z}Pi{x, y}Pi{x}Pi{y}Pi{z}Pi{Φ},
which is not additive since {^}Fj{y} and {y, z}Pi{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.



More intriguing information

1. Effort and Performance in Public-Policy Contests
2. Cyclical Changes in Short-Run Earnings Mobility in Canada, 1982-1996
3. The name is absent
4. The demand for urban transport: An application of discrete choice model for Cadiz
5. The name is absent
6. ISO 9000 -- A MARKETING TOOL FOR U.S. AGRIBUSINESS
7. The name is absent
8. Family, social security and social insurance: General remarks and the present discussion in Germany as a case study
9. The name is absent
10. Evolutionary Clustering in Indonesian Ethnic Textile Motifs