Voting by Committees under Constraints

3.3 Separable Preferences

Theorem 2 below characterizes the family of non-dictatorial and strategy-
proof social choice functions when voters’ preferences are separable. Our
result shows that the class of strategy-proof social choice functions under ad-
ditive representable preferences identified in Theorem 1 is drastically reduced
as a consequence of this enlargement of the domain of preferences. Again,
this is a novelty with respect to the situation without constraints. Now,
only social choice functions with Cartesian product ranges (up to constant
and/or omitted objects,) are strategy-proof. Namely, the range of F has to
be a subcube: all sections of the minimal Cylindric decomposition of ⅛ (the
set of not omitted objects) are singletons, either with the object itself as the
unique active component (constant object) or else with the object itself and
the empty set as the two active components. Formally,

Theorem 2 Let F: Sⁿ 2^κ be a non-dictatorial social choice function
with ffR_F ≥ 3. Then_r F is strategy-proof if and only if F is voting by
committees and all sections of the minimal cylindric decomposition of R_f
are singletons.

Proof Let F: S^,' -→ 2^fζ' bf' a non-dictatorial social choice function with
Φ-R_f ≥ 3. If a social choice function is voting by committees and the sec-
tions of the minimal cylindric decomposition of R_f are all singletons, then
this social choice function is onto the power set of some subset of the objects
union a constant disjoint set. The result of Barberà, Sonnenschein, and Zhou
(1991) applies, and the function is strategy-proof. For the converse, assume
that F is strategy-proof. By Proposition 2, F is voting by committees. Since
all additive preferences are separable, Theorem 1 applies to the subdomain
of additive preferences. Therefore, the committees associated to F satisfy

18

<<   16   17   18   19   20   21   22   >>

More intriguing information