# Gibbard–Satterthwaite theorem | sketch of proof

## Sketch of proof

The Gibbard–Satterthwaite theorem can be proved based on Arrow's impossibility theorem, which deals with social ranking functions, i.e. voting systems designed to yield a complete preference order of the candidates, rather than simply choosing a winner. We give a sketch of proof in the simplified case where the voting rule ${\displaystyle f}$ is assumed to be unanimous. It is possible to build a social ranking function ${\displaystyle \operatorname {Rank} }$, as follows: in order to decide whether ${\displaystyle a\prec b}$, the ${\displaystyle \operatorname {Rank} }$ function creates new preferences in which ${\displaystyle a}$ and ${\displaystyle b}$ are moved to the top of all voters' preferences. Then, ${\displaystyle \operatorname {Rank} }$ examines whether ${\displaystyle f}$ chooses ${\displaystyle a}$ or ${\displaystyle b}$. It is possible to prove that, if ${\displaystyle f}$ is non-manipulable and non-dictatorial, then ${\displaystyle \operatorname {Rank} }$ satisfies the properties: unanimity, independence of irrelevant alternatives, and it is not a dictatorship. Arrow's impossibility theorem says that, when there are three or more alternatives, such a ${\displaystyle \operatorname {Rank} }$ function cannot exist. Hence, such an ${\displaystyle f}$ function also cannot exist.[7]:214–215