Gibbard–Satterthwaite theorem

The Gibbard–Satterthwaite theorem is a theorem in voting theory. It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 and then proved independently by the philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner, and states that for every voting rule of this form, at least one of the following three things must hold:

1. The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
2. The rule limits the possible outcomes to two alternatives only; or
3. The rule is susceptible to tactical voting: in some situations, a voter's sincere ballot may not best defend their opinion.

The scope of this theorem is limited to ordinal voting. It does not apply to cardinal voting systems such as score voting or STAR voting, nor does it apply to multi-winner voting, nondeterministic methods or non-voting decision mechanisms.

Gibbard's theorem is more general and covers processes of collective decision that may not be ordinal, such as cardinal voting. Gibbard's 1978 theorem and Hylland's theorem are even more general and extend these results to non-deterministic processes, where the outcome may depend partly on chance; the Duggan–Schwartz theorem extends these results to multiwinner electoral systems.