Dictatorial Voting Rule (COMSOC)

Definition

Dictatorial Voting Rule ( COMSOC)

A voting rule is dictatorial if one fixed voter $d$, called the dictator, always determines the outcome through their first-ranked alternative. The same voter must do so on every profile; being decisive in one election is not dictatorship.

Write $R=({\succ }_1,\dots ,{\succ }_n)$ for the submitted profile and $\mathrm{top}({\succ }_d)$ for the first alternative in voter $d$‘s ranking.

Single-winner output

For a rule $f$ returning one alternative,

$$\exists d\in \{1,\dots ,n\}\forall R\in \mathcal{L}(C{)}^n:f(R)=\mathrm{top}({\succ }_d).$$

Other voters’ reports cannot change the winner while the dictator’s first choice stays fixed.

Set-valued output

Under the same sole-winner convention, a rule $\lambda$ satisfies

$$\exists d\in \{1,\dots ,n\}\forall R\in \mathcal{L}(C{)}^n:\lambda (R)=\{\mathrm{top}({\succ }_d)\}.$$

Every output is a singleton containing the dictator’s first choice. Merely requiring that choice to belong to $\lambda (R)$ would be weaker: it would allow other co-winners and would not let the voter determine the entire outcome.