Editing Gibbard–Satterthwaite theorem (section)

== Sketch of proof ==
The Gibbard–Satterthwaite theorem can be proved using [[Arrow's impossibility theorem]] for [[Social ranking function|social ranking functions]]. We give a sketch of proof in the simplified case where some voting rule <math>f</math> is assumed to be [[Pareto-efficient]]. 

It is possible to build a social ranking function <math>\operatorname{Rank}</math>, as follows: in order to decide whether <math>a\prec b</math>, the <math>\operatorname{Rank}</math> function creates new preferences in which <math>a</math> and <math>b</math> are moved to the top of all voters' preferences.{{Clarify|date=March 2024}} Then, <math>\operatorname{Rank}</math> examines whether <math>f</math> chooses <math>a</math> or <math>b</math>.

It is possible to prove that, if <math>f</math> is non-manipulable and non-dictatorial, <math>\operatorname{Rank}</math> satisfies independence of irrelevant alternatives. Arrow's impossibility theorem says that, when there are three or more alternatives, such a <math>\operatorname{Rank}</math> function must be a [[Dictatorship mechanism|dictatorship]]. Hence, such a voting rule <math>f</math> must also be a dictatorship.<ref name="agt">{{Cite Algorithmic Game Theory 2007}}</ref>{{rp|214–215}}

Later authors have developed other variants of the proof.<ref name="reny" /><ref name="benoit" /><ref name="sen" /><ref name="gardenfors">{{cite journal |last1=Gärdenfors |first1=Peter |date=1977 |title=A Concise Proof of Theorem on Manipulation of Social Choice Functions |journal=Public Choice |volume=32 |pages=137–142 |doi=10.1007/bf01718676 |issn=0048-5829 |jstor=30023000 |s2cid=153421058}}</ref><ref name="barbera">{{cite journal |last1=Barberá |first1=Salvador |date=1983 |title=Strategy-Proofness and Pivotal Voters: A Direct Proof of the Gibbard-Satterthwaite Theorem |journal=International Economic Review |volume=24 |issue=2 |pages=413–417 |doi=10.2307/2648754 |issn=0020-6598 |jstor=2648754}}</ref><ref>{{cite book |last=Dummett |first=Michael |title=Voting Procedures |publisher=Oxford University Press |year=1984 |isbn=978-0198761884}}</ref><ref>{{cite journal |last1=Fara |first1=Rudolf |last2=Salles |first2=Maurice |date=2006 |title=An interview with Michael Dummett: From analytical philosophy to voting analysis and beyond |url=http://eprints.lse.ac.uk/552/1/VPP05_01.pdf |journal=Social Choice and Welfare |volume=27 |issue=2 |pages=347–364 |doi=10.1007/s00355-006-0128-9 |jstor=41106783 |s2cid=46164353}}</ref><ref>{{cite book |last1=Moulin |first1=Hervé |url=http://www.cambridge.org/il/academic/subjects/economics/microeconomics/axioms-cooperative-decision-making |title=Axioms of Cooperative Decision Making |publisher=Cambridge University Press |year=1991 |isbn=9780521424585 |author-link=Hervé Moulin |access-date=10 January 2016}}</ref><ref>{{cite journal |last=Taylor |first=Alan D. |date=April 2002 |title=The manipulability of voting systems |journal=[[The American Mathematical Monthly]] |volume=109 |issue=4 |pages=321–337 |doi=10.2307/2695497 |jstor=2695497}}</ref>{{Excessive citations inline|date=March 2024}}