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 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. Arrow's impossibility theorem says that, when there are three or more alternatives, such a function cannot exist. Hence, such an function also cannot exist.[7]:214–215