Resolute Social Choice Correspondence (COMSOC)

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.