高级微观经济学博弈论讲义

StaticGamesofCompleteInformation-Lecture1Dec,2006,FudanUniversityGameTheory--Lecture12OutlineofStaticGamesofCompleteInformationIntroductiontogamesNormal-form(orstrategic-form)representationIteratedeliminationofstrictlydominatedstrategiesNashequilibriumApplicationsofNashequilibriumMixedstrategyNashequilibriumDec,2006,FudanUniversityGameTheory--Lecture13AgendaWhatisgametheoryExamplesPrisoner’sdilemmaThebattleofthesexesMatchingpenniesStatic(orsimultaneous-move)gamesofcompleteinformationNormal-formorstrategic-formrepresentationDec,2006,FudanUniversityGameTheory--Lecture14Whatisgametheory?Wefocusongameswhere:ThereareatleasttworationalplayersEachplayerhasmorethanonechoicesTheoutcomedependsonthestrategieschosenbyallplayers;thereisstrategicinteractionExample:Sixpeoplegotoarestaurant.Eachpersonpayshis/herownmeal–asimpledecisionproblemBeforethemeal,everypersonagreestosplitthebillevenlyamongthem–agameDec,2006,FudanUniversityGameTheory--Lecture15Whatisgametheory?Gametheoryisaformalwaytoanalyzestrategicinteractionamongagroupofrationalplayers(oragents)GametheoryhasapplicationsEconomicsPoliticsSociologyLawetc.Dec,2006,FudanUniversityGameTheory--Lecture16ClassicExample:Prisoners’DilemmaTwosuspectsheldinseparatecellsarechargedwithamajorcrime.However,thereisnotenoughevidence.Bothsuspectsaretoldthefollowingpolicy:Ifneitherconfessesthenbothwillbeconvictedofaminoroffenseandsentencedtoonemonthinjail.Ifbothconfessthenbothwillbesentencedtojailforsixmonths.Ifoneconfessesbuttheotherdoesnot,thentheconfessorwillbereleasedbuttheotherwillbesentencedtojailforninemonths.-1,-1-9,0

0,-9-6,-6Prisoner1Prisoner2ConfessMumConfessMumDec,2006,FudanUniversityGameTheory--Lecture17Example:ThebattleofthesexesAttheseparateworkplaces,ChrisandPatmustchoosetoattendeitheranoperaoraprizefightintheevening.BothChrisandPatknowthefollowing:Bothwouldliketospendtheeveningtogether.ButChrispreferstheopera.Patpreferstheprizefight.2,10,00,01,2ChrisPatPrizeFightOperaPrizeFightOperaDec,2006,FudanUniversityGameTheory--Lecture18Example:MatchingpenniesEachofthetwoplayershasapenny.TwoplayersmustsimultaneouslychoosewhethertoshowtheHeadortheTail.Bothplayersknowthefollowingrules:Iftwopenniesmatch(bothheadsorbothtails)thenplayer2winsplayer1’spenny.Otherwise,player1winsplayer2’spenny.-1,11,-11,-1-1,1Player1Player2TailHeadTailHeadDec,2006,FudanUniversityGameTheory--Lecture19Static(orsimultaneous-move)gamesofcompleteinformationAsetofplayers(atleasttwoplayers)Foreachplayer,asetofstrategies/actionsPayoffsreceivedbyeachplayerforthecombinationsofthestrategies,orforeachplayer,preferencesoverthecombinationsofthestrategies{Player1,Player2,...Playern}S1S2...Snui(s1,s2,...sn),forall

s1S1,s2S2,...snSn.Astatic(orsimultaneous-move)gameconsistsof:Dec,2006,FudanUniversityGameTheory--Lecture110Static(orsimultaneous-move)gamesofcompleteinformationSimultaneous-moveEachplayerchooseshis/herstrategywithoutknowledgeofothers’choices.Completeinformation(ongame’sstructure)Eachplayer’sstrategiesandpayofffunctionarecommonknowledgeamongalltheplayers.AssumptionsontheplayersRationalityPlayersaimtomaximizetheirpayoffsPlayersareperfectcalculatorsEachplayerknowsthatotherplayersarerationalDec,2006,FudanUniversityGameTheory--Lecture111Static(orsimultaneous-move)gamesofcompleteinformationTheplayerscooperate?No.Onlynon-cooperativegamesMethodologicalindividualismThetimingEachplayerichooseshis/herstrategysi

withoutknowledgeofothers’choices.Theneachplayerireceiveshis/herpayoff

ui(s1,s2,...,sn).Thegameends.Dec,2006,FudanUniversityGameTheory--Lecture112Definition:normal-formorstrategic-formrepresentationThenormal-form(orstrategic-form)representationofagameGspecifies:Afinitesetofplayers{1,2,...,n},players’strategyspacesS1S2...Snandtheirpayofffunctionsu1u2...un

whereui:S1×S2×...×Sn→R.Dec,2006,FudanUniversityGameTheory--Lecture113Normal-formrepresentation:2-playergameBi-matrixrepresentation2players:Player1andPlayer2EachplayerhasafinitenumberofstrategiesExample:

S1={s11,s12,s13}S2={s21,s22}Player2s21s22Player1s11u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)s13u1(s13,s21),u2(s13,s21)u1(s13,s22),u2(s13,s22)Dec,2006,FudanUniversityGameTheory--Lecture114Classicexample:Prisoners’Dilemma:

normal-formrepresentationSetofplayers: {Prisoner1,Prisoner2}Setsofstrategies:

S1

=S2

={Mum,Confess}Payofffunctions:

u1(M,M)=-1,u1(M,C)=-9,u1(C,M)=0,u1(C,C)=-6;

u2(M,M)=-1,u2(M,C)=0,u2(C,M)=-9,u2(C,C)=-6-1,-1-9,00,-9-6,-6Prisoner1Prisoner2ConfessMumConfessMumPlayersStrategiesPayoffsDec,2006,FudanUniversityGameTheory--Lecture115Example:ThebattleofthesexesNormal(orstrategic)formrepresentation:Setofplayers: {Chris,Pat}(={Player1,Player2})Setsofstrategies: S1

=S2={Opera,PrizeFight}Payofffunctions:

u1(O,O)=2,u1(O,F)=0,u1(F,O)=0,u1(F,O)=1;

u2(O,O)=1,u2(O,F)=0,u2(F,O)=0,u2(F,F)=22,10,00,01,2ChrisPatPrizeFightOperaPrizeFightOperaDec,2006,FudanUniversityGameTheory--Lecture116Example:MatchingpenniesNormal(orstrategic)formrepresentation:Setofplayers: {Player1,Player2}Setsofstrategies: S1

=S2={Head,Tail}Payofffunctions:

u1(H,H)=-1,u1(H,T)=1,u1(T,H)=1,u1(H,T)=-1;

u2(H,H)=1,u2(H,T)=-1,u2(T,H)=-1,u2(T,T)=1-1,11,-11,-1-1,1Player1Player2TailHeadTailHeadDec,2006,FudanUniversityGameTheory--Lecture117Example:Tourists&NativesOnlytwobars(bar1,bar2)inacityCanchargepriceof$2,$4,or$56000touristspickabarrandomly4000nativesselectthelowestpricebarExample1: Bothcharge$2eachgets5,000customersand$10,000Example2: Bar1charges$4,Bar2charges$5Bar1gets3000+4000=7,000customersand$28,000Bar2gets3000customersand$15,000Dec,2006,FudanUniversityGameTheory--Lecture118Example:CournotmodelofduopolyAproductisproducedbyonlytwofirms:firm1andfirm2.Thequantitiesaredenotedbyq1andq2,respectively.Eachfirmchoosesthequantitywithoutknowingtheotherfirmhaschosen.ThemarketpriceisP(Q)=a-Q,whereQ=q1+q2.ThecosttofirmiofproducingquantityqiisCi(qi)=cqi.Thenormal-formrepresentation:Setofplayers: {Firm1,Firm2}Setsofstrategies: S1=[0,+∞),S2=[0,+∞)Payofffunctions:

u1(q1,q2)=q1(a-(q1+q2)-c),u2(q1,q2)=q2(a-(q1+q2)-c)Dec,2006,FudanUniversityGameTheory--Lecture119OneMoreExampleEachofnplayersselectsanumberbetween0and100simultaneously.Letxi

denotethenumberselectedbyplayeri.LetydenotetheaverageofthesenumbersPlayeri’spayoff=xi–3y/5Thenormal-formrepresentation:Dec,2006,FudanUniversityGameTheory--Lecture120SolvingPrisoners’DilemmaConfessalwaysdoesbetterwhatevertheotherplayerchoosesDominatedstrategyThereexistsanotherstrategywhichalwaysdoesbetterregardlessofotherplayers’choices-1,-1-9,00,-9-6,-6Prisoner1Prisoner2ConfessMumConfessMumPlayersStrategiesPayoffsDec,2006,FudanUniversityGameTheory--Lecture121Definition:strictlydominatedstrategy-1,-1-9,00,-9-6,-6Prisoner1Prisoner2ConfessMumConfessMumregardlessofotherplayers’choicessi”isstrictlybetterthansi’Dec,2006,FudanUniversityGameTheory--Lecture122ExampleTwofirms,ReynoldsandPhilip,sharesomemarketEachfirmearns$60millionfromitscustomersifneitherdoadvertisingAdvertisingcostsafirm$20millionAdvertisingcaptures$30millionfromcompetitorPhilipNoAdAdReynoldsNoAd60,6030,70Ad70,3040,40Dec,2006,FudanUniversityGameTheory--Lecture1232-playergamewithfinitestrategiesS1={s11,s12,s13}S2={s21,s22}s11

isstrictlydominatedby

s12

if

u1(s11,s21)<u1(s12,s21)and

u1(s11,s22)<u1(s12,s22).s21isstrictlydominatedby

s22

if

u2(s1i,s21)<u2(s1i,s22),fori=1,2,3Player2s21s22Player1s11u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)s13u1(s13,s21),u2(s13,s21)u1(s13,s22),u2(s13,s22)Dec,2006,FudanUniversityGameTheory--Lecture124Definition:weaklydominatedstrategy1,12,00,22,2Player1Player2RUBLregardlessofotherplayers’choicessi”isatleastasgoodassi’Dec,2006,FudanUniversityGameTheory--Lecture125StrictlyandweaklydominatedstrategyArationalplayerneverchoosesastrictlydominatedstrategy.Hence,anystrictlydominatedstrategycanbeeliminated.Arationalplayermaychooseaweaklydominatedstrategy.Theorderofeliminationdoesnotmatterforstrictdominanceelimination(pindownthesameequilibrium),butdoesforweakone.Dec,2006,FudanUniversityGameTheory--Lecture126IteratedeliminationofstrictlydominatedstrategiesIfastrategyisstrictlydominated,eliminateitThesizeandcomplexityofthegameisreducedEliminateanystrictlydominatedstrategiesfromthereducedgameContinuedoingsosuccessivelyDec,2006,FudanUniversityGameTheory--Lecture127Iteratedeliminationofstrictlydominatedstrategies:anexample1,01,20,10,30,12,0Player1Player2MiddleUpDownLeft1,01,20,30,1Player1Player2MiddleUpDownLeftRightDec,2006,FudanUniversityGameTheory--Lecture128Example:Tourists&NativesOnlytwobars(bar1,bar2)inacityCanchargepriceof$2,$4,or$56000touristspickabarrandomly4000nativesselectthelowestpricebarExample1: Bothcharge$2eachgets5,000customersand$10,000Example2: Bar1charges$4,Bar2charges$5Bar1gets3000+4000=7,000customersand$28,000Bar2gets3000customersand$15,000Dec,2006,FudanUniversityGameTheory--Lecture129Example:Tourists&NativesBar2$2$4$5Bar1$210,1014,1214,15$412,1420,2028,15$515,1415,2825,25PayoffsareinthousandsofdollarsBar2$4$5Bar1$420,2028,15$515,2825,25Dec,2006,FudanUniversityGameTheory--Lecture130OneMoreExampleEachofnplayersselectsanumberbetween0and100simultaneously.Letxi

denotethenumberselectedbyplayeri.LetydenotetheaverageofthesenumbersPlayeri’spayoff=xi–3y/5Dec,2006,FudanUniversityGameTheory--Lecture131OneMoreExampleThenormal-formrepresentation:Players:{player1,player2,...,playern}Strategies:Si

=[0,100],fori=1,2,...,n.Payofffunctions:ui(x1,x2,...,xn)=xi–3y/5Isthereanydominatedstrategy?Whatnumbersshouldbeselected?Dec,2006,FudanUniversityGameTheory--Lecture132Newsolutionconcept:NashequilibriumPlayer2LCRPlayer1T0,44,05,3M4,00,45,3B3,53,56,6Thecombinationofstrategies(B,R)hasthefollowingproperty:Player1CANNOTdobetterbychoosingastrategydifferentfromB,giventhatplayer2choosesR.Player2CANNOTdobetterbychoosingastrategydifferentfromR,giventhatplayer1choosesB.Dec,2006,FudanUniversityGameTheory--Lecture133Newsolutionconcept:NashequilibriumPlayer2L’C’R’Player1T’0,44,03,3M’4,00,43,3B’3,33,33.5,3.6Thecombinationofstrategies(B’,R’)hasthefollowingproperty:Player1CANNOTdobetterbychoosingastrategydifferentfromB’,giventhatplayer2choosesR’.Player2CANNOTdobetterbychoosingastrategydifferentfromR’,giventhatplayer1choosesB’.Dec,2006,FudanUniversityGameTheory--Lecture134NashEquilibrium:ideaNashequilibriumAsetofstrategies,oneforeachplayer,suchthateachplayer’sstrategyisbestforher,giventhatallotherplayersareplayingtheirequilibriumstrategiesDec,2006,FudanUniversityGameTheory--Lecture135Definition:NashEquilibriumGivenothers’choices,playericannotbebetter-offifshedeviatesfromsi*(cf:dominatedstrategy)Prisoner2MumConfessPrisoner1Mum-1,-1-9,0Confess0,-9-6,-6Dec,2006,FudanUniversityGameTheory--Lecture1362-playergamewithfinitestrategiesS1={s11,s12,s13}S2={s21,s22}(s11,

s21)isaNashequilibriumif

u1(s11,s21)

u1(s12,s21),

u1(s11,s21)

u1(s13,s21)and

u2(s11,s21)

u2(s11,s22).Player2s21s22Player1s11u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)s13u1(s13,s21),u2(s13,s21)u1(s13,s22),u2(s13,s22)Dec,2006,FudanUniversityGameTheory--Lecture137FindingaNashequilibrium:cell-by-cellinspection1,01,20,10,30,12,0Player1Player2MiddleUpDownLeft1,01,20,30,1Player1Player2MiddleUpDownLeftRightDec,2006,FudanUniversityGameTheory--Lecture138Example:Tourists&NativesBar2$2$4$5Bar1$210,1014,1214,15$412,1420,2028,15$515,1415,2825,25PayoffsareinthousandsofdollarsBar2$4$5Bar1$420,2028,15$515,2825,25Dec,2006,FudanUniversityGameTheory--Lecture139OneMoreExampleThenormal-formrepresentation:Players:{player1,player2,...,playern}Strategies:Si

=[0,100],fori=1,2,...,n.Payofffunctions:ui(x1,x2,...,xn)=xi–3y/5WhatistheNashequilibrium?Dec,2006,FudanUniversityGameTheory--Lecture140Bestresponsefunction:exampleIfPlayer2choosesL’thenPlayer1’sbeststrategyisM’IfPlayer2choosesC’thenPlayer1’sbeststrategyisT’IfPlayer2choosesR’thenPlayer1’sbeststrategyisB’IfPlayer1choosesB’thenPlayer2’sbeststrategyisR’Bestresponse:thebeststrategyoneplayercanplay,giventhestrategieschosenbyallotherplayersPlayer2L’C’R’Player1T’0,44,03,3M’4,00,43,3B’3,33,33.5

,3.6Dec,2006,FudanUniversityGameTheory--Lecture141Example:Tourists&NativeswhatisBar1’sbestresponsetoBar2’sstrategyof$2,$4or$5?whatisBar2’sbestresponsetoBar1’sstrategyof$2,$4or$5?Bar2$2$4$5Bar1$210,1014,1214,15$412,1420,2028,15$515,1415,2825,25PayoffsareinthousandsofdollarsDec,2006,FudanUniversityGameTheory--Lecture1422-playergamewithfinitestrategiesS1={s11,s12,s13}S2={s21,s22}Player1’sstrategys11

isherbestresponsetoPlayer2’sstrategy

s21

if

u1(s11,s21)

u1(s12,s21)and

u1(s11,s21)

u1(s13,s21).Player2s21s22Player1s11u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)s13u1(s13,s21),u2(s13,s21)u1(s13,s22),u2(s13,s22)Dec,2006,FudanUniversityGameTheory--Lecture143UsingbestresponsefunctiontofindNashequilibriumIna2-playergame,(s1,s2)isaNashequilibriumifandonlyifplayer1’sstrategys1isherbestresponsetoplayer2’sstrategys2,andplayer2’sstrategys2isherbestresponsetoplayer1’sstrategys1.-1,-1-9,0

0

,-9-6

,-6Prisoner1Prisoner2ConfessMumConfessMumDec,2006,FudanUniversityGameTheory--Lecture144UsingbestresponsefunctiontofindNashequilibrium:exampleM’isPlayer1’sbestresponsetoPlayer2’sstrategyL’

T’isPlayer1’sbestresponsetoPlayer2’sstrategyC’

B’isPlayer1’sbestresponsetoPlayer2’sstrategyR’

L’isPlayer2’sbestresponsetoPlayer1’sstrategyT’C’isPlayer2’sbestresponsetoPlayer1’sstrategyM’

R’isPlayer2’sbestresponsetoPlayer1’sstrategyB’Player2L’C’R’Player1T’0,44,03,3M’4,00,43,3B’3,33,33.5

,3.6Dec,2006,FudanUniversityGameTheory--Lecture145Example:Tourists&NativesBar2$2$4$5Bar1$210,1014,1214,15$412,1420,2028,15$515,1415,2825,25PayoffsareinthousandsofdollarsUsebestresponsefunctiontofindtheNashequilibrium.Dec,2006,FudanUniversityGameTheory--Lecture146Example:ThebattleofthesexesOperaisPlayer1’sbestresponsetoPlayer2’sstrategyOperaOperaisPlayer2’sbestresponsetoPlayer1’sstrategyOperaHence,(Opera,Opera)isaNashequilibriumFightisPlayer1’sbestresponsetoPlayer2’sstrategyFightFightisPlayer2’sbestresponsetoPlayer1’sstrategyFightHence,(Fight,Fight)isaNashequilibrium

2

,

10,00,0

1

,

2ChrisPatPrizeFightOperaPrizeFightOperaDec,2006,FudanUniversityGameTheory--Lecture147Example:MatchingpenniesHeadisPlayer1’sbestresponsetoPlayer2’sstrategyTailTailisPlayer2’sbestresponsetoPlayer1’sstrategyTailTailisPlayer1’sbestresponsetoPlayer2’sstrategyHeadHeadisPlayer2’sbestresponsetoPlayer1’sstrategyHeadHence,NONashequilibrium-1,

1

1

,-1

1

,-1-1,

1Player1Player2TailHeadTailHeadDec,2006,FudanUniversityGameTheory--Lecture148Definition:bestresponsefunctionPlayeri’sbestresponseGiventhestrategieschosenbyotherplayersDec,2006,FudanUniversityGameTheory--Lecture149Definition:bestresponsefunctionPlayeri’sbestresponsetootherplayers’strategiesisanoptimalsolutiontoDec,2006,FudanUniversityGameTheory--Lecture150UsingbestresponsefunctiontodefineNashequilibriumAsetofstrategies,oneforeachplayer,suchthateachplayer’sstrategyisbestforher,giventhatallotherplayersareplayingtheirstrategies,orAstablesituationthatnoplayerwouldliketodeviateifotherssticktoitDec,2006,FudanUniversityGameTheory--Lecture151CournotmodelofduopolyAproductisproducedbyonlytwofirms:firm1andfirm2.Thequantitiesaredenotedbyq1andq2,respectively.Eachfirmchoosesthequantitywithoutknowingtheotherfirmhaschosen.ThemarketpricedisP(Q)=a-Q,whereaisaconstantnumberandQ=q1+q2.ThecosttofirmiofproducingquantityqiisCi(qi)=cqi.Dec,2006,FudanUniversityGameTheory--Lecture152CournotmodelofduopolyThenormal-formrepresentation:Setofplayers: {Firm1,Firm2}Setsofstrategies: S1=[0,+∞),S2=[0,+∞)Payofffunctions:

u1(q1,q2)=q1(a-(q1+q2)-c)

u2(q1,q2)=q2(a-(q1+q2)-c)Dec,2006,FudanUniversityGameTheory--Lecture153CournotmodelofduopolyHowtofindaNashequilibriumFindthequantitypair(q1*,q2*)suchthatq1*isfirm1’sbestresponsetoFirm2’squantityq2*andq2*isfirm2’sbestresponsetoFirm1’squantityq1*Thatis,q1*solves

Maxu1(q1,q2*)=q1(a-(q1+q2*)-c)

subjectto0q1+∞

andq2*solves

Max

u2(q1*,q2)=q2(a-(q1*+q2)-c)

subjectto0q2+∞Dec,2006,FudanUniversityGameTheory--Lecture154CournotmodelofduopolyHowtofindaNashequilibriumSolve

Maxu1(q1,q2*)=q1(a-(q1+q2*)-c)

subjectto0q1+∞

FOC:a-2q1-q2*-c=0

q1=(a-q2*-c)/2Dec,2006,FudanUniversityGameTheory--Lecture155CournotmodelofduopolyHowtofindaNashequilibriumSolve

Max

u2(q1*,q2)=q2(a-(q1*+q2)-c)

subjectto0q2+∞

FOC:a-2q2–q1*–

c=0

q2=(a–q1*–

c)/2Dec,2006,FudanUniversityGameTheory--Lecture156CournotmodelofduopolyHowtofindaNashequilibriumThequantitypair(q1*,q2*)isaNashequilibriumif

q1*=(a–q2*–

c)/2

q2*=(a–q1*–

c)/2Solvingthesetwoequationsgivesus

q1*=q2*=(a–

c)/3Dec,2006,FudanUniversityGameTheory--Lecture157CournotmodelofduopolyBestresponsefunctionFirm1’sbestfunctiontofirm2’squantityq2:

R1(q2)=(a–q2

–

c)/2ifq2<a–

c;0,othwise

Firm2’sbestfunctiontofirm1’squantityq1:

R2(q1)=(a–q1

–

c)/2ifq1<a–

c;0,othwiseq1q2(a–

c)/2(a–

c)/2a–

ca–

cNashequilibriumDec,2006,FudanUniversityGameTheory--Lecture158CournotmodelofoligopolyAproductisproducedbyonlynfirms:firm1tofirmn.Firmi’squantityisdenotedbyqi.Eachfirmchoosesthequantitywithoutknowingtheotherfirms’choices.ThemarketpricedisP(Q)=a-Q,whereaisaconstantnumberandQ=q1+q2+...+qn.ThecosttofirmiofproducingquantityqiisCi(qi)=cqi.Dec,2006,FudanUniversityGameTheory--Lecture159CournotmodelofoligopolyThenormal-formrepresentation:Setofplayers: {Firm1,...Firmn}Setsofstrategies: Si=[0,+∞),fori=1,2,...,nPayofffunctions:

ui(q1,...,qn)=qi(a-(q1+q2+...+qn)-c)

fori=1,2,...,nDec,2006,FudanUniversityGameTheory--Lecture160CournotmodelofoligopolyHowtofindaNashequilibriumFindthequantities(q1*,...qn*)suchthatqi*isfirmi’sbestresponsetootherfirms’quantitiesThatis,q1*solves

Maxu1(q1,q2*,...,qn*)=q1(a-(q1+q2*+...+qn*)-c)

subjectto0q1+∞

andq2*solves

Max

u2(q1*,q2,q3*,...,qn*)=q2(a-(q1*+q2+q3*+...+qn*)-c)

subjectto0q2+∞

.......Dec,2006,FudanUniversityGameTheory--Lecture161Bertrandmodelofduopoly(differentiatedproducts)Twofirms:firm1andfirm2.Eachfirmchoosesthepriceforitsproductwithoutknowingtheotherfirmhaschosen.Thepricesaredenotedbyp1andp2,respectively.Thequantitythatconsumersdemandfromfirm1:q1(p1,p2)=a

–

p1+bp2.Thequantitythatconsumersdemandfromfirm2:q2(p1,p2)=a

–

p2+bp1.ThecosttofirmiofproducingquantityqiisCi(qi)=cqi.Dec,2006,FudanUniversityGameTheory--Lecture162Bertrandmodelofduopoly(differentiatedproducts)Thenormal-formrepresentation:Setofplayers: {Firm1,Firm2}Setsofstrategies: S1=[0,+∞),S2=[0,+∞)Payofffunctions:

u1(p1,p2)=(a

–

p1+bp2)(p1

–

c)

u2(p1,p2)=(a

–

p2+bp1)(p2

–

c)Dec,2006,FudanUniversityGameTheory--Lecture163Bertrandmodelofduopoly(differentiatedproducts)HowtofindaNashequilibriumFindthepricepair(p1*,p2*)suchthatp1*isfirm1’sbestresponsetoFirm2’spricep2*andp2*isfirm2’sbestresponsetoFirm1’spricep1*Thatis,p1*solves

Maxu1(p1,p2*)=(a

–

p1+bp2*)(p1

–

c)

subjectto0p1+∞

andp2*solves

Max

u2(p1*,p2)=(a

–

p2+bp1*)(p2

–

c)

subjectto0p2+∞Dec,2006,FudanUniversityGameTheory--Lecture164Bertrandmodelofduopoly(differentiatedproducts)HowtofindaNashequilibriumSolvefirm1’smaximizationproblem

Maxu1(p1,p2*)=(a

–

p1+bp2*)(p1

–

c)

subjectto0p1+∞

FOC:a+c–2p1

+bp2*

=0

p1=(a+c+bp2*)/2Dec,2006,FudanUniversityGameTheory--Lecture165Bertrandmodelofduopoly(differentiatedproducts)HowtofindaNashequilibriumSolvefirm2’smaximizationproblem

Max

u2(p1*,p2)=(a

–

p2+bp1*)(p2

–

c)

subjectto0p2+∞

FOC:a+c–2p2

+bp1*

=0

p2=(a+c+bp1*)/2Dec,2006,FudanUniversityGameTheory--Lecture166Bertrandmodelofduopoly(differentiatedproducts)HowtofindaNashequilibriumThepricepair(p1*,p2*)isaNashequilibriumif

p1*=(a+c+bp2*)/2

p2*=(a+c+bp1*)/2Solvingthesetwoequationsgivesus

p1*=p2*=(a+

c)/(2–b)Dec,2006,FudanUniversityGameTheory--Lecture167Bertrandmodelofduopoly(homogeneousproducts)Twofirms:firm1andfirm2.Eachfirmchoosesthepriceforitsproductwithoutknowingtheotherfirmhaschosen.Thepricesaredenotedbyp1andp2,respectively.Thequantitythatconsumersdemandfromfirm1:q1(p1,p2)=a

–

p1ifp1<p2;=(a

–

p1)/2

ifp1=p2;=0,

ow.Thequantitythatconsumersdemandfromfirm2:q2(p1,p2)=a

–

p2ifp2<p1;=(a

–

p2)/2

ifp1=p2;=0,

ow.ThecosttofirmiofproducingquantityqiisCi(qi)=cqi.Dec,2006,FudanUniversityGameTheory--Lectur