Voting by Committees under Constraints

(3) ¾4 is dictatorial and equal for all x’s in the same section in Rp s min-
imal cylindric decomposition, when this section has more than two active
components.

The proof of Theorem 1 is in the Appendix at the end of the paper. We
discuss here an elaborate example which illustrates almost all the possibilities
which are opened by our characterization.

Example 3 Let K = {a,b,x,y, z,w,r, s,q,t} be the set of objects and
assume that the set of feasible alternatives is

{{br} , {brt} , {bsq} , {bsqt} , {bzr} , {bzrt} , {bzsq} , {bzsqt} , {bzwr} ,
{bzwrt} , {bzwsq} , {bzwsqt} , {bxr} , {bxrt} , {bxsq} , {bxsqt} ,
{bxzr} , {bxzrt} , {bxzsq} , {bxzsqt} , {bxzwr} , {bxzwrt} , {bxzwsq} ,
{bxzwsqt} , {byr} , {byrt} , {bysq} , {bysqt} , {byzr} , {byzrt} ,
{byzsq} , {byzsqt} , {byzwr} , {byzwrt} , {byzwsq} , {byzwqst}}.

Notice that (1) a is never chosen, (2) b is always chosen, (3) x and y are never
chosen simultaneously, (4) w is only chosen if z is, (5) s and q are chosen
together, (6) exactly one of r and s is chosen, and (7) t can be chosen or
not, whatever happens. Therefore, we are interested in strategy-proof social
choice functions F: Aⁿ → 2^κ whose range is equal to

R_f = {_b} ₊ {0, {4 , {y}} ₊ {0, {_z} , {z, w}} ₊ {{r} , {_s, q}} ₊ {0, {t}} .

Notice that the partition {{b} , {x, y} , {z, w} , {r, s, q} , {t}} of K is the mini-
mal cylindric decomposition of Rf, since one can check that all of its elements
are minimal sections. For example, {x, y} is a section because AC ({x, y}) =
{0, {^},{y}} (notice that the subset {rr,y} is not an active component of
itself) and C∖F∙^,'> (0), C^_f^,v^ ({z}), and C∖F∙^,'> ({y}) are all equal to

{b} + {0, {z} , {A w}} + {{r} , {_s, q}} ₊ {0, {t}} .

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