## Gibbard–Satterthwaite theorem | sketch of proof |

- informal description
- formal statement
- examples
- corollary
- sketch of proof
- history
- posterity
- see also
- references

The Gibbard–Satterthwaite theorem can be proved based on **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 is assumed to be unanimous. It is possible to build a social ranking function , as follows: in order to decide whether , the function creates new preferences in which and are moved to the top of all voters' preferences. Then, examines whether chooses or . It is possible to prove that, if is non-manipulable and non-dictatorial, then satisfies the properties: unanimity, independence of irrelevant alternatives, and it is not a dictatorship. ^{[7]}^{:214–215}