08 Nash equilibrium location segregation and randomization.docx
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08 Nash equilibrium location segregation and randomization.docx
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08Nashequilibriumlocationsegregationandrandomization
GameTheory:
Lecture8Transcript
Overview:
Wefirstcompleteourdiscussionofthecandidate-votermodelshowing,inparticular,that,inequilibrium,twocandidatescannotbetoofarapart.ThenweplayandanalyzeSchelling'slocationgame.Wediscusshowsegregationcanoccurinsocietyevenifnoonedesiresit.Wealsolearnthatseeminglyirrelevantdetailsofamodelcanmatter.Weconsiderrandomizationsfirstbyacentralauthority(suchasinabussingpolicy),andthendecentralizedrandomizationbytheindividualsthemselves,"mixedstrategies."Finally,welookatrock,paper,scissorstoseeanexampleofamixed-strategyequilibriumtoagame.
October1,2007
< 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
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