耶鲁大学公开课博弈论3.docx
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耶鲁大学公开课博弈论3.docx
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耶鲁大学公开课博弈论3
耶鲁大学公开课_博弈论3
GameTheory:
Lecture3Transcript
September12,2007< ProfessorBenPolak: Lasttimewetalkedalotabouttheidea--Ididn'tmentionthenamebuthere'ssomejargonforyou--wetalkedabouttheideaofdeletingdominatedstrategies: lookingatagame;figuringoutwhichstrategiesaredominated;deletingthem;lookingatthegameagain;lookingatwhichstrategiesarenowdominated;deletingthose;andsoonandsoforth.Thisprocessiscalledthe"iterativedeletionofdominatedstrategies."ThisisnotagreattitleforBarry'sbook.Youneedtodobetterthanthattogetthetwohundredandfiftydollars.Thisisaboringtitle,butnevermind. IterativeDeletionofDominatedStrategies.Theideais--Itembodiestheideaofputtingyourselfinsomeoneelse'sshoesandtryingtofigureoutwhatthey'regoingtodo,andthenthinkaboutthemputtingthemselvesinyourshoes,figuringoutwhatyou'regoingtodo,andsoonandsoforth.Lasttime,wesawalready,thatthisisaverypowerfulidea,inthatgamelasttime.Butwealsosawit'sadangerousideatotaketooliterally: thatsometimesthiscangetyoutoover-thinktheproblemandactually,asinthatnumbersgamelasttime,thebestchoice,thewinningchoice,mightnotinvolvesomanyrounds. Wealsosawinthatnumbersgamelasttimethatinsomegames,butbynomeansallgames,insomegamesthisprocessactuallyconvergestoasinglechoice.Inthatnumbersgameitconvergedto1.Soonceagainwhatthisideais--thisideais--youyourselfshouldnotplayadominatedstrategy.Deletethose.Deletethoseforeveryoneelse,becauseeveryoneelseisnotgoingtoplayadominatedstrategy.Deletethose.Lookatthegamewithallthosedominatedstrategiesdeleted.Seeifthereareanystrategiesthatarenowdominated.Deletethose.Lookagain,etc.,etc.Icouldwritethatallup.Wesawitinpracticelasttime.It'sprobablyjusteasiertohavetheideathere. Onetipaboutthis,trytoidentifyallthedominatedstrategiesofallplayersbeforeyoudelete,thendelete.Thenlookagain.Trytoidentifyallthedominatedstrategiesofallplayersagain,andthendelete.Thatprocesswillpreventyoufromgettingintotrouble.Sotoday,Iwanttobealittlebitlessabstract,ifyoulike,andIwanttolookatanapplication.Iwanttolookatafamousapplicationfrompolitics. Sowe'regoingtolookatamodelofpolitics.Theideaisgoingtobethis.We'regoingtoimaginethattherearetwocandidates,andwhatthesecandidatesaredoingisthey'rechoosingtheirpoliticalpositionsforanelection. Sothesearetheplayers,andthestrategiesaregoingtobe-they'regoingtochoosepositionsonaspectrum,onapoliticalspectrum.Tomakelifeeasy,we'regoingto assumethatthispoliticalspectrumhastenpositions.Soherearethepositions.We'llcallthem1,2,3,4,5,6,7,8,9,and10.Verydifficult.What'stheideahere? Wehaven'tfinisheddescribingthegame.What'stheidea? Theideaisthatthesepositionsareleftwingpositionsandthesepositionsarerightwingpositions.(Canpeoplenotseepastthepodium? Let'sgetitoutoftheway.) Soyoucouldthinkoftheseextremeleftwingpositions.Thesearepeoplewho,Iguess,theydon'teatanythingexceptforfruitandtheythinkthattreesshouldhavethevote.Theseguysouthere,thesearetheextremerightwingpositions.Sotheythinkthepoorshouldn'thavethevoteandtheyeatimmigrants.I'manimmigrant: betterbecareful.Thecandidatesherearegoingtotryandchoosepositions.We'regoingtoassume--thisisnotrealistic--we'regoingtoassumefornowthatthereare10%ofthevotersateachofthesepositions.Sothere's10%ofthevotersateachposition.Sotheyareuniformlydistributed. We'regoingtoassumethatvoterswilleventuallyvotefortheclosestcandidate.Sovotersvotefortheclosestcandidate: thecandidatewhosepositionisclosesttotheirown.Andwe'llneedatiebreakingassumptionandwe'lldotheobvioustiebreak.Ifthere'satiethenthevoterssplit.Thevotersofthatpositionsplitevenly.Ifthere'satie,halfthevotersatthatpositiongoforoneofthecandidatesandhalfofthemgofortheother.Sohere'sagame,I'vegottheplayers,that'sthecandidates.I'vegotthestrategies,that'sthepoliticalpositions.WhatamImissing? I'mmissingpayoffs,right? I'mmissingpayoffs.Itmattersinthisgamealothowwespecifythepayoffsbutwe'regoingtoassumethatthepayoffsarethatthecandidatesaimtomaximizetheirshareofthevote. Butthat'snottheonlythingwecouldhaveassumed.Wecouldhaveassumedthatalltheyreallycareaboutiswinningandthatwinninggavethemahighpayoffandthatlosinggavethemnothing.I'mgoingtoassumealittlebitmoreandI'mgoingtoassumethat--Ihaven'tspelledmaximizedright,maxwilldo--theywanttomaxtheirshareofthevote.Thereasonthisisn'taterribleassumptionisyoucouldthinkthatgettingahighershareofthevotegivesyouamandate.Or,ifthisisaprimaryelection,alargershareofthevotegivesyouabiggerpushforthenextprimaryorwhatever.Sowe'llassumethatthey'retryingtomaximizetheirshareofthevote. Sowewanttoknowwhat'sgoingtohappeninthisgame.Itseemslikeaprettynatural,prettyimportantgame.Okay,sogivenwhatwe'velearnedsofarintheclass,anaturalfirstquestiontoaskis: areanyofthestrategieshere? Therearetenstrategies1,2,3,4,5,6,7,8,9,10,areanyofthestrategiesdominated? Areanystrategiesdominated? Let'sgetsome--Letmegetmymicrophonesupandreadyalittlebit.Soanystrategiesdominated? Howaboutthisgentlemaninblue? Sostandupandshout,yeah. Student: Okay,1and10arebothdominated. ProfessorBenPolak: Sothisgentlemanwhosenameis? Student: Steven. ProfessorBenPolak: Stevensays1and10.We'llcomebackto10.You'reright.We'llcomebackto10.Sohesaysthatposition1,strategy1,choosingthemostextremeleftwingpositionisadominatedstrategy.Whatdominatesitbytheway? Steven,youwanttoshoutitout? Student: 2. ProfessorBenPolak: 2.Soforexample--Soourconjecturehereisthat2dominates1.That'stheclaim.Let'sjustbecarefulhere,whatdoesitmeantosaythat2dominates1? Itmeansthatchoosingposition2alwaysgivesmeahighershareofthevotethanchoosingposition1,nomatterwheretheothercandidatepositionsherself.Itdoesnotmean2beats1. Solet'stakeSteven'sconjectureandseeifit'strue.So,inparticular,let'sstartworkingoutwhatshareofthevotesyou'dgetifyouchoseposition1orposition2,againstdifferentpositionstheotherguycanchoose.So,forexample,whatwe'regoingtodoiswe'regoingtotestdoes2dominate1? Whilewe'redoingthiswe'llfigureouthowthesepayoffsworkaswell.Well,howaboutversus1.Sosupposetheothercandidatehaschosenposition1.Sotheothercandidatehaschosenposition1.Thenwe'regoingtocomparemypayofffromchoosing1againsttheothercandidate'spayofffromchoosing1.We'regoingtocomparethiswithmypayofffromchoosing2againsttheothercandidatechoosing1. Let'sfigureoutwhatthatis.Sowhat'smy--somebodycanshoutthisout--what'smyshareofthevoteifIchoose1andtheothercandidatechooses1? 50%.Thatwasprettyeasy,right? Itmustbeatieforeverybody,soI'llget50%ofthevote.What'smyshareofthevoteifIchoose2andtheothercandidatechooses1? 90%.Hewillgetorshewillgetallthevotersat1.I'llgeteveryoneelse.SoIget90%.Sointhiscasechoosing2isbetterthanchoosing1.Butofcoursewe'renotdoneyet.Wehavetoconsiderotherpossiblepositionsofmyopponent. Sosupposemyopponentchooses2.Sonowwe'recomparingmychoosing1,whenmyopponentchooses2,andthepayoffIwouldgetifIchoose2whenmyopponentchooses2.Sowhat'smypayoffifIchoose1andmyopponentchooses2? Igetallthevotersrightontopofmeatposition1andshegetseveryoneelse,isthatright? SoIget10%.Andwhataboutifwebothchoose2? 50%,everybodyhappywiththat? So50%inthiscase,andonceagain,2didbetterthan1.Everyonehappywiththat? Soweshowedthischoicewasbetterandthischoicewasbetter,andwe'regoingtokeepongoing. Against3: sonowwe'recomparingmychoosing1versus3andmychoosing2versus3.IfIchoose1against3,whatshareofthevotedoIget? I'mgoingtogetallthepeopleatposition1andhalfthepeopleatposition2foratotalof15.AndifIchose2against3,whatdoIget? Iget20,Igetallthepeopleat1andIgetallthepeople2,soIget20%,sowe'reokayagain.Let'sdoonemorejusttoseeapatternhere.SoifIchoose1against4,andcomparingmychoosing2against4.IntheformercaseifIchoose1against4,howmanyvotesdoIget? Igetallthepeopleat1,andallthepeopleat2,soIget20%ofthevotes,shegetseveryoneelse.AndhereifIchoose2against4,Igetallthepeopleat1,allthepeopleat2,andwhat? Halfthepeopleat3,sothatcomesoutas25%andonceagainandsoon. Now,Icouldgoonrightthewaythroughhere,butI'mgoingtostopbecauseitgetstobeabitboringafterawhile.I'mguessingyoucanseeapatternemerginghere.What'sthepatternhere? What'sthepatternhere? Somebody? Where'sthat? Somebodyraisetheirhandsowecangetamickeoutthere.Canwegetamikeonthisguyinwhite? Standup,standup,yep,standupandshout. Student: 2isalwaysbetterthan1. ProfessorBenPolak: 2isalwaysbetterthan1,butwecanseemorethanthatinthepattern.That'sright,butwhat'sthepatternhere? Mikebackhere. Student: Whatever'scloserto5isbetterthanwhatever'sfart
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