Voting by Committees under Constraints

separable ones. Section 4 contains a final remark. Section 5 contains the
proof of Theorem 1, omitted in Section 3.

2 Preliminaries

Agents are the elements of a finite set N = {1,2, ...,n}. The set of objects
is K = {1,...,⅛}. We assume that n and к are at least 2. Generic elements
of N will be denoted by i and j and generic elements of K will be denoted
by x, y, and z. Alternatives are subsets of K which will be denoted by X,
Y, and Z. Subsets of N will be represented by I and J. Calligraphic letters
will represent families of subsets; for instance, X. У, and Z will represent
families of subsets of alternatives and TV, ɪ, and J families of subsets of
agents (coalitions).

Preferences are binary relations on alternatives. Let P be the set of
complete, transitive, and asymmetric preferences on 2^κ. Preferences in P
are denoted by P_i, Pj, P'_i, and F∙. For P_i ∈P and X Ç 2^κ, we denote the
alternative in X most-preferred according to P_i as τχ (Pf), and we call it the
top of P_i on X. We will use τ (Pf) to denote the top of P_i on 2^κ. Generic
subsets of preferences will be denoted by P.

Preference profiles are n-tuples of preferences. They will be represented
by F = (Fl,..., P_n) or by F= (P_i, P_f) if we want to stress the role of agents
i^,s preference.

A social choice function on P is a function F: P" → 2^κ.

Definition 1 The social choice function F : Pⁿ → 2^κ respects voter’s
sovereignty if for every X E 2^κ there exists P ∈P" such that F(P) = X.

The range of a social choice function F: Pⁿ —> 2^κ is denoted by ''Rp : that

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