Arrow’s Theorem and the Gibbard-Satterthwaite

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1、Arrow’sTheoremandtheGibbard-SatterthwaiteTheorem:AUni…edApproach¤PhilipJ.RenyDepartmentofEconomicsUniversityofChicagoFirstVersion:December1999ThisVersion:June2000AbstractTheconnectionbetweenArrow’stheoremandtheGibbard-Satterthwaitetheoremisfurtherstrengthenedbyprovidingasingleproofthatyieldsbothre

2、sults.Keywords:Arrow’sTheorem,Gibbard-SatterthwaiteTheorem.JELClassi…cationNumberD71.1.ASharedProofLetAdenotea…nitesetofalternativesandletLdenotethesetofstrictlinearorders,or(strict)rankings,onA:LetL¤denotethesetofweaklinearorders,or(weak)rankings,onA:FixapositiveintegerN:Afunctionf:LN!Awillbecall

3、edasocialchoicefunction,whileafunctionF:LN!L¤willbecalledasocialwelfarefunction.AmemberofLNiscalledapro…leofrankings(orsimplyapro…le)anditsithcomponentiscalledindividuali’sranking.AmemberofL¤iscalledasocialorder,orsociety’sranking.Wesaythatasocialchoicefunctionf:LN!Ais:ParetoE¢cientifwheneveralter

4、nativeaisatthetopofeveryindividuali’sranking,Li;thenf(L1;:::;LN)=a.Monotonicifwheneverf(L1;:::;LN)=aandforeveryindividualiandeveryalter-nativebtherankingL0ranksaabovebifLdoes,thenf(L0;:::;L0)=a.ii1N¤IwishtothankVijayKrishnaandMottyPerryformotivatingthepresentnotethroughastimulatingdiscussionofasho

5、rtproofofArrow’sTheoremduetoJohnGeanakoplos.ThanksalsotoDrewFudenbergandEricMaskinforhelpfulcommentsandtoMattJacksonandRogerMyersonforbringingtomyattentiontheworkofMullerandSatterthwaite(1977),toHugoSonnenscheinfordirectingmetotheworkofBarberà(1980,1983)andtoMarkSatterthwaiteforremindingmeofthewor

6、kofSchmeidlerandSonnenschein(1978).FinancialsupportfromthetheNationalScienceFoundation(SBR-9709392andSES-9905599)isgratefullyacknowledged.Dictatorialifthereisanindividualisuchthatf(L1;:::;LN)=aifandonlyifaisatthetopofi’srankingLi:WesaythatasocialwelfarefunctionF:LN!L¤is:ParetoE¢cientifwheneveralte

7、rnativeaisrankedabovebaccordingtoeachLi;thenaisrankedabovebaccordingtoF(L1;:::;LN):IndependentofIrrelevantAlternatives(IIA)ifwhenevertherankingofaversusbisunchangedforeachi=1;:::;Nwhenindividuali’srankingchangesf