Voting by Committees under Constraints

Moreover, this section is minimal since neither {ж} nor {y} are sections
because, for instance, AC ({ж}) = {0, {ж}} but

Φ (в) = {ц + {в, ω} + {в, {₂} .m) + {{>■}. o∙. «}} + {», w}
and

({≈}) = {6} ₊ {0,{z} . {z,w}} + {{r} . {_i>,<∕}} + {0. {«}} ,

and hence, C'j'' (0) ≠ C, ^,∙!^ι'})

Also, {^,w} is a section because AC ({z, w}) = {0, {z} , {z, w}} (notice
that the subset {w} is not an active component of {z, w}) and CfC^w'^s (0),
^,w} and ρAA       _{are a}n equal to

W + {0,W,M} + {W,⅛Q}} + {0,{t}}∙

Moreover, this section is minimal since neither {z} nor {w} are sections
because, for instance, AC ({w}) = {0, { w } } but

4"¹ (0) = {!>} + {0. Ы . {y}} + {0. {z}} + {{r} . {_i>,<∕}} + {0. {«}}
and

4”’ (W) = {4 + {0. w. ⅛}} + w + {{и, {∙_s,₉}} + {0, и,

and hence, C^ (0) ≠ C^ ({w}).

The proof that all other components of the decomposition are also mini-
mal sections is similar and left to the reader.

Now, given a set of agents N, any voting by committees /■': A" → 2^λ
will be strategy-proof as long as it satisfies the following properties: (a) by
condition (3) of Theorem 1, W''' = W(" = {¼}} and TV™ = W™ = {{⅛}}
for some i_i,i₂ ∈ N; (b) by condition (1) of Theorem 1, W('' = TV™; and (c)
by condition (2) of Theorem 1, TV_r and TV_s are complementary.

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