Surjective Voting Rule (COMSOC)

Definition

Surjective Voting Rule ( COMSOC)

A voting rule is surjective in the alternative-wise sense if every alternative wins for at least one profile. “Has a chance to win” means that a suitable profile exists, not that the rule uses randomness.

Single-winner output

For $f:\mathcal{L}(C{)}^n\to C$,

$$\forall x\in C\exists R\in \mathcal{L}(C{)}^n:f(R)=x.$$

This is ordinary surjectivity onto $C$. A singleton-valued representation $\lambda (R)=\{f(R)\}$ gives the equivalent condition $\lambda (R)=\{x\}$.

Set-valued output

For $\lambda :\mathcal{L}(C{)}^n\to 2^C\setminus \{\varnothing \}$, the analogous requirement is

$$\forall x\in C\exists R\in \mathcal{L}(C{)}^n:x\in \lambda (R).$$

Every alternative can be a co-winner. This is not surjectivity onto the full set-valued codomain: not every non-empty subset of $C$ must occur as an output.