Editing Gibbard–Satterthwaite theorem (section)

{{Short description|Impossibility result for ranked-choice voting systems}}{{Expert needed|game theory
| date = June 2024
| reason = inadequate description of theorem and practical importance
}}
{{Use dmy dates|date=August 2017}}The '''Gibbard–Satterthwaite theorem''' is a theorem in [[social choice theory]]. It was first conjectured by the philosopher [[Michael Dummett]] and the mathematician [[Robin Farquharson]] in 1961<ref>{{cite journal|author=Rudolf Farra and Maurice Salles|title=An Interview with Michael Dummett: From analytical philosophy to voting analysis and beyond|journal=Social Choice and Welfare|volume=27|issue=2|date=October 2006|doi=10.1007/s00355-006-0128-9|pages=347–364|s2cid=46164353|url=http://eprints.lse.ac.uk/552/1/VPP05_01.pdf}}</ref> and then proved independently by the philosopher [[Allan Gibbard]] in 1973<ref name="gibbard">{{cite journal
|first=Allan
|last=Gibbard
|author-link=Allan Gibbard
|title=Manipulation of voting schemes: A general result
|journal=Econometrica
|volume=41
|issue=4
|year=1973
|pages=587–601
|jstor=1914083
|doi=10.2307/1914083
}}</ref> and economist [[Mark Satterthwaite]] in 1975.<ref name="satterthwaite">{{cite journal
|first=Mark Allen
|last=Satterthwaite
|author-link=Mark Satterthwaite
|title=Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions
|journal=Journal of Economic Theory
|volume=10
|issue=2
|date=April 1975
|pages=187–217
|doi=10.1016/0022-0531(75)90050-2
|citeseerx=10.1.1.471.9842
}}</ref> It deals with deterministic [[Ranked voting|ordinal electoral system]]s that choose a single winner, and shows that for every voting rule of this form, at least one of the following three things must hold:

# The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
# The rule limits the possible outcomes to two alternatives only; or
# The rule is not straightforward, i.e. there is no single [[Strategic dominance|always-best strategy]] (one that does not depend on other voters' preferences or behavior).
[[Gibbard's theorem|Gibbard's proof of the theorem]] is more general and covers processes of collective decision that may not be ordinal, such as [[cardinal voting]].{{NoteTag|Gibbard's theorem does not imply that cardinal methods necessarily incentivize reversing one's relative rank of two candidates.}} [[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.