information:filtering
Sketch of proof

The Gibbard–Satterthwaite theorem can be proved using Arrow's impossibility theorem for social ranking functions. We give a sketch of proof in the simplified case where some voting rule f {\displaystyle f} is assumed to be Pareto-efficient.

It is possible to build a social ranking function Rank {\displaystyle \operatorname {Rank} }, as follows: in order to decide whether a ≺ b {\displaystyle a\prec b}, the Rank {\displaystyle \operatorname {Rank} } function creates new preferences in which a {\displaystyle a} and b {\displaystyle b} are moved to the top of all voters' preferences. Then, Rank {\displaystyle \operatorname {Rank} } examines whether f {\displaystyle f} chooses a {\displaystyle a} or b {\displaystyle b}.

It is possible to prove that, if f {\displaystyle f} is non-manipulable and non-dictatorial, Rank {\displaystyle \operatorname {Rank} } satisfies independence of irrelevant alternatives. Arrow's impossibility theorem says that, when there are three or more alternatives, such a Rank {\displaystyle \operatorname {Rank} } function must be a dictatorship. Hence, such a voting rule f {\displaystyle f} must also be a dictatorship.

Later authors have developed other variants of the proof.