ProfessorBenPolak:
Solasttimeweleftthingsinthemiddleofamodel,whichwasthecandidate-votermodel.Whatwas--Idon'twanttogooverthewholemodelagain,butjusttoreiteratealittlebit--whatwasdifferentaboutthatmodelfromwhatwesawbefore--themainthingthatwasdifferentwasthatthecandidatescannotchoosetheirpositions.Ifyoulike,everyvoterisapotentialcandidatebutyouknowthepositionsofthevoters.
Letmejustbringouttwolessonsthatwelefthanginglasttime.Iwanttojustputthemontheboardtomakesurethey'reinyournotes.Sothefirstlessonis--onethingwesawalreadylasttime--istherecanbelotsofdifferentNashEquilibriumthismodel.TherearemultiplepossibleNashEquilibriuminthismodelandmoretothepoint,notallofthoseequilibriahavethecandidatescrowdedatthecenter.WesawearlyonintheclassicDownsormedian-votermodelthatthatmodelpredictedcrowdingthecenter.Thisonedoesn't,andwe'llcomebacktothatinasecond.Andasecondthingwesawlasttimewasthatentrycan--ifyouenterontheleftoneaffectofenteringontheleftcanbetomakethecandidateontherightmorelikelytowin.Conversely,ifyouenterontheright--you'rearight-wingvotercandidateandyouenter,thatcanmakeitmorelikelythattheleftwingsare--canleadtothewinnerbeingmore--beingfurtherfromyouridealposition.
Justtorevisitthisalittlebit,letmegobackandjustillustratethosetwopointsagain.SoI'lltakearowfurtherbackthistimeandgetanicefullrowthatwecanseethewholeof.Thisway,I'mgoingtotakearowfurtherbacksothatthistimewehavenoconfusionaboutwhatleftwingandrightwingis,atleastalmostnoconfusion.Soletmechoosethisrow,thisrowgoodokay.I'msorry,thepeopleinthebalconyaregoingtohavetoimaginethis.It'sapenaltybeinginthebalcony.Sothisrow.Andhere'smyleftwingofthisrow（foreveryonewho'sinfrontofyouwhichisalmosteverybody）,andhere'smyrightwing.
Let'stryandillustratesomeequilibriumwesawlasttime.So,inparticular,ifIcangettheguyintheblueYaleshirttostandupasecondandtheguywithhiscomputertostandupasecond.Sorry.Let'sassumealltheseatsarefilledfornowso--justtohelpmeoutalittlebitsinceI'mdoingthisonthefly.This,I'mgoingtoclaimisanequilibrium.Noticethattherearetwocandidatesstandingandnoticethattheyarenotparticularlyclosetothecenter.Wehaveoursortofmiddle-of-the-democratic-partyleftcandidates.SoI'mtemptedtogiveyouanamebutperhapsIwillnot.Andhere'samiddle-of-the-republican-partycandidate.Andthisistheelection.They'regoingtosplitthevotesequallyifI'veactuallychosencorrectly.
So,justtoobservesomethingshere.First,forthistobeanequilibrium,itbetterbethecasethatthey'resymmetriconleftandright.Ifthey'renotsymmetric,thenitisn'tanequilibriumbecauseoneofthosecandidatesisgoingtoloseforsureandthewaywesetupthemodelthatmeansthatoneofthemwilldropout:
theywilldeviatetodropout.Isthatcorrect?
Let'salsoillustratethis.So,we'vealreadyillustratedthatthey'renotparticularlyclosetothecenter.Let'sillustratewhatImeantbysomebodyontheleftcausingtherightwingtowin.So,ifthiswastheelection,herewearegettingcloserandclosertotheelection,andsuddenlyoneofourleft-wingguys,soDennisKucinichorsomethingdecidestoenter--solet'ssupposethisisDennisKucinichandheenters.IfDennisKucinich,ourleft-wingguyenters,theremightbesomesortofmoralvictoryinentering,buttheresultofthiswillbethatourright-wingcandidatewins;everyoneseesthatnow?
Ifthesethreeguysarestanding,DennisKucinichisgoingtostealsomevotesfromourcenter-leftcandidateandcauseourcenter-rightcandidatetowin.Now,thatmayormaynotbeMr.Kucinich'sintention,butweshouldatleastbeawareofit.So,thisisarealeffect.Gobacktothe2000electionandthinkaboutwhathappenedwhenNadarenteredand"dothemath"asitwere,whichpeopledidn'tdoatthetimeapparently.IfyouwerelisteningtothenewspapersorifyoureadTheNewYorkTimesthismorningorlistenedtotheradiothismorning,you'llfindexactlythesamedebateisgoingonnowintherepublicanparty.Somemember--actuallyit'stheotherwayaround,it'stherightwingthistime;Isupposeweshouldswitchitaround.
Sotheright--somepeopleontherightoftherepublicanpartyaresayingthatifitturnsoutthatGiuliani,whocurrentlyisleadinginthepolls,winstherepublicannominationtheywillrunathird-partycandidate.Ofcoursethedebateisactuallyintheseterms.Iftheyrunaright-wingthird-partycandidate--sothatwouldbeourguyoverhere,thismightbe--thismighthavepayoffintermsofotherthings.Butintermsoftheelection,it'sgoingtoresultinHilarywinning.Sowe'veseenthesetwoeffects.They'reveryrealeffects.We'renotnecessarilygettingpeoplecrowdingthecenterandwehavetoworryabout,whenweenter,causingtheotherwingtowin.Thisisn't--I'mnotmakingaleftwing/rightwingpoliticsargumenthere--thisistrueforbothwingssymmetrically.
So,let'stryandbringonemoreideainhere,whichiswhereweendeduplasttime,whichisjusthowfarawayfromthecentercanwebe?
Sotheseguyssitdownasecondandlet'sstandup,Mr.Kucinich,again;Iknowthatisn'tyourrealnamebutnevermind.Mr.KucinichouthereontheleftandMr.crazy-right-wingguywhosenameI've--who'sthemostcrazyrightwingguyofthesecandidates?
I'mnot--I'llgetintroublewhoeverIname--sowhoeverthemostcrazyright-wingcandidateyoucanthinkof.andnowwehavethefullspectrumrepresentedwithjusttheextremesstanding.Herewehave--they'resymmetricaroundthecenter,butIclaimthisisnotanequilibrium.Sothepeopleinthebalcony,I'vegottheextremerightandtheextremeleftstandinghere.Whyisthisnotanequilibrium?
YeahIshould--it'smyfaultIshouldhavebroughtthemike.CanIhavethemike?
I'msorrymyfault.Thanks.Thisone.SoKatie,whyisthatnotanequilibrium?
Student:
Becausethepersoninthecentercouldstandupandwinthemajority.
ProfessorBenPolak:
Exactly,becausethepersoninthecentercouldstandupandwin.Actually,itdoesn'tonlyhavetobethepersonexactlyinthecenter.Awidearrayofcentercandidatescoulddeviateandwinatthispoint.So,ifthiswasthetwocandidatesstanding,andyouimagineathirdcandidatestanding,whoforexample,isthisgentleman,ifhewastostand,wecould--hehastobealittlebitclosertothecenter,let'ssaythisguy,theguyingray--fairlyclearlyhe'sgoingtoendupwinning.Sothisisthethirdlesson.Ifthecandidatesaretoofarapartwe'regoingtoseesomecenterentry,whichisgoingtowin.Thankyouguys.Soeventhoughthereisn'tthisfullDownsianeffectofpushingcandidatestowardsthecenter,eventhoughwedon'thavethemedian-votertheoremhere,westillhavepartoftheintuitionsurviving.Thepartoftheintuitionthat'ssurvivesis,ifthecandidatesaretoofarapart,thenthecenterwillenterandwin.Sothereissomethingpullingpeopletothecenterstill.
Soareasonablequestionhereisjusthowfarapartinequilibriumcanthecandidatesbe?
We'veestablishedthatwecanhavetwocandidatesandtheyneedn'tbothbeatthecenter.We'veestablishedthattheycan'tbeattheextremes.Howfarapartcantheybe?
Well,thisisreallyjusta--it'skindofanerdyquestiontogetitpreciselybutlet'sgetitpreciselynevertheless.
Solet'shavealook,let'susetheotherboard.So,here'sthefullextentofourpoliticalspectrumfrom0to1,andletmejusttryandillustratehowfarapartthesecanbe.WhatI'mgoingtodoisI'mgoingtodividethisintosixths-1/6,2/6,1/2,4/6,5/6.SoIclaim--andI'llshowafterwards--I'llclaimthatprovidedthetwocandidatesaren'toutsideof1/6and5/6thatthatwillbeanequilibrium.So,inparticular,ifthecandidatesarejustinside1/6andjustinside5/6,orjustmorethan1/6andjustlessthan5/6--sohere'soneofthesecandidateswho'sstandingandhere'stheotherone--thenwe'llbeokay.Nowwhy?
Whyisthattherightanswer?
Anyonewanttotryaguess?
ThiswasthequestionIsentyouovertheweekend.I'msureyouweredoingotherthingsovertheweekendbutnevertheless,whyisthisanswer?
Well,let'ssee.Whataretheyvulnerabletoo?
They'revulnerabletodeviationbysomebodyenteringatthecenter.Andifsomebodyentersatthecenter--whatwouldmakesomebodyenteratthecenter?
They'regoingtoenteratthecenteriftheycanwinbasically.Soiftheyenteratthecenterinthiscase,let'sseehowmanyvoteseveryonegets.Soifweenteratthecenterhere--here'sournewcandidate--who'sthinkingaboutenteringatthecenter.Sohe'ssortofthinkingaboutit.Sowhat'shisorhercalculationgoingtobe?
Well,let'slookatwha
升级会员 