Winner (COMSOC)

Definition

Winner ( COMSOC)

A winner is an alternative selected by a voting rule for a given profile $R$.

Set-valued rule

For a social choice correspondence $\lambda$,

$$c\text{ is a winner}⟺c\in \lambda (R).$$

This is also called a co-winner. The winner is unique exactly when $\lambda (R)=\{c\}$.

Single-valued rule

For a social choice function $f$,

$$c\text{ is the winner}⟺f(R)=c.$$

The rule selects exactly one winner for each profile.