Social Choice Function (COMSOC)

Definition

Social Choice Function ( COMSOC)

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

$$f:\underset{\text{one ranking per voter}}{\underbrace{\mathcal{L}(C{)}^n}}⟶\underset{\text{one winning alternative}}{\underbrace{C}}.$$

Here $\mathcal{L}(C)$ is the set of preference rankings. For every profile $R$, the rule selects exactly one winner $f(R)\in C$, unlike a social choice correspondence, which returns a non-empty set of winners.

A tie-breaking rule can turn a correspondence $\lambda$ into a function by selecting one alternative $f(R)\in \lambda (R)$ for each profile.