Social Choice Correspondence (COMSOC)

Definition

Social Choice Correspondence ( COMSOC)

For a non-empty set of alternatives $C$ and $n$ voters, a social choice correspondence is a voting rule

$$\lambda :\underset{\text{one ranking per voter}}{\underbrace{\mathcal{L}(C{)}^n}}⟶\underset{\text{non-empty sets of winners}}{\underbrace{2^C\setminus \{\varnothing \}}}.$$

Here $\mathcal{L}(C)$ is the set of preference rankings, and $2^C$ is the power set of $C$. Given a profile $R$, the rule returns a non-empty set $\lambda (R)\subseteq C$ of winning alternatives. Several winners are allowed; no winner is not.

Unlike a social choice function, it need not select a single winner. Multiple winners do not necessarily form an elected committee; they may be tied alternatives.

Resoluteness

Definition

Resolute Social Choice Correspondence ( COMSOC)

A social choice correspondence $\lambda :\mathcal{L}(C{)}^n\to 2^C\setminus \{\varnothing \}$ is resolute if it selects exactly one winner for every preference profile:

$$\forall R\in \mathcal{L}(C{)}^n:|\lambda (R)|=1.$$

It therefore determines a social choice function $f:\mathcal{L}(C{)}^n\to C$ by

$$\underset{\text{singleton winner set}}{\underbrace{\lambda (R)=\{f(R)\}}}.$$

Resoluteness concerns the whole voting rule: a unique winner on just one profile is not enough.

Link to original