Social choice function ($f:{\mathcal{R}}^n\to A$)

A deterministic social choice function maps profiles of preference orders to single alternatives. Unrestricted domain permits every profile in ${\mathcal{R}}^n$; onto means every alternative lies in the range. A unanimous rule selects an alternative whenever every agent ranks it first.

With an odd number of agents and two alternatives, select the alternative ranked first by a strict majority. The rule is onto, strategyproof, and for at least three agents has no dictator in social choice. A pivotal agent already gets its preferred alternative by reporting truthfully; a nonpivotal agent cannot change the outcome alone.

A dictatorship always selects the top alternative of one fixed agent $d$. Other agents cannot alter the selection against that agent's preference. This is the social-choice meaning of dictatorship, rather than a political classification.

A dictator is the fixed agent whose top-ranked alternative is always selected by a dictatorship in social choice.