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GameTheory--
Lecture6
PatrickLoiseauEURECOMFall2016
1
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Outline
1. Stackelberg duopolyandthefirstmover’sadvantage
2. Formaldefinitions3. Bargaininganddiscountedpayoffs
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Outline
1. Stackelberg duopolyandthefirstmover’sadvantage
2. Formaldefinitions3. Bargaininganddiscountedpayoffs
3
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Cournot Competitionreminder• Theplayers:2Firms,e.g.CokeandPepsi• Strategies:quantitiesplayersproduceofidenticalproducts:qi,q-i– Productsareperfectsubstitutes
• Thepayoffs– Constantmarginalcostofproductionc– Marketclearingprice:p=a– b(q1 +q2)– firmsaimtomaximizeprofit
u1(q1,q2)=p*q1 – c*q1
0
a
q1 +q2
pSlope:-b
Demandcurve
4
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Nashequilibrium
• u1(q1,q2)=a*q1 – b*q21 – b*q1 q2 – c*q1• FOC,SOCgivebestresponses:
• NEiswhentheycross:
à Cournot quantity 5
ïïî
ïïí
ì
--
==
--
==
22)(ˆ
22)(ˆ
1122
2211
qbcaqBRq
qbcaqBRq
bcaqq
qbcaq
bca
qqqBRqBR
3
2222
)()(
*2
*1
12
*2
*11221
-==Þ
--
=--
=Þ=
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Graphically
6
0 q1
q2
bca
2-
bca -
NEMonopoly
Perfectcompetition
BR2
BR1
bcaqCournot 3
-=
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Stackelberg Model• Assumenowthatonefirmgetstomovefirstandtheothermovesafter– Thatisonefirmgetstosetthequantityfirst
• Isitanadvantagetomovefirst?– Oritisbettertowaitandseewhattheotherfirmisdoingandthenreact?
• Wearegoingtousebackwardinductiontocomputethequantities– Wecannotdrawtreesherebecauseofthecontinuumofpossibleactions
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Intuition
• Suppose1movesfirst• 2respondsbyBR!(bydef)• Whatquantityshouldfirm1produce,knowingthatfirm2willrespondusingtheBR?– constrainedoptimizationproblem
80 q1
q2
BR2
q’1q’’1
q’2
q’’2
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Intuition(2)• Shouldfirm1producemoreorlessthantheCournotquantity?– Productsarestrategicsubstitutes:themorefirm1produces,thelessfirm2willproduceandvice-versa
– Firm1producingmoreè firm1ishappy• Whathappenstofirm1’sprofits?– Theygoup,otherwisefirm1wouldn’thavesethigherproductionquantities
• Whathappenstofirm2’sprofits?– Theanswerisnotimmediate
• Whathappenedtothetotaloutputinthemarket?– Evenheretheanswerisnotimmediate
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Intuition(3)
• Whathappenedtothetotaloutputinthemarket?– Consumerswouldlikethetotaloutputtogoup,forthatwouldmeanthatpriceswouldgodown!
– Indeed,itgoesdown:seetheBRcurve
10
0 q1
q2
BR2
q’’1q’1
q’’2q’2
Theincrementfromq’1toq’’1islargerthanthedecrementfromq’2toq’’2
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Intuition(4)
• Whathappenstofirm2’sprofits?– q1wentup,q2wentdown– q1+q2wentupè priceswentdown– Firm2’scostsarethesame
èFirm2’sprofitwentdown
• Wehaveseenthatfirm1’sprofitgoesup
èConclusion:Firstmoverisanasset(here!) 11
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Stackelberg Modelcomputations
• Letusnowcomputethequantities.Wehave
• WeapplytheBackwardInductionprinciple– First,solvethemaximizationproblemforfirm2,takingq1asgiven
– Then,focusonfirm1
€
p = a − b(q1 + q2)profit i = pqi − cqi
12
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Stackelberg Modelcomputations(2)
• Firm2’soptimizationproblem(forfixedq1)
• Wenowcantakethisquantityandplugitinthemaximizationproblemforfirm1
( )[ ]
22
max
12
2
22212
qbcaq
q
cqqbqbqaq
--
=Þ¶¶
---
13
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Stackelberg Modelcomputations(3)
• Firm1’soptimizationproblem:
€
maxq1
a − bq1 − bq2( )q1 − cq1[ ] =
maxq1
a − bq1 − ba − c2b
−q12
#
$ %
&
' (
#
$ %
&
' ( − c
)
* +
,
- . q1 =
maxq1
a − c2
−bq12
)
* + ,
- . q1 =max
q1
a − c2
q1 − bq12
2)
* +
,
- .
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Stackelberg Modelcomputations(4)
• WederiveF.O.C.andS.O.C.
• Thisgivesus
0
02
0
21
2
11
<-=¶¶
=--
Þ=¶¶
bq
bqcaq
bca
bca
bcaq
bcaq
4221
2
2
2
1
-=
--
-=
-=
15
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Stackelberg quantities
• Allthismathtoverifyourinitialintuition!
CournotNEW
CournotNEW
qqqq
22
11
<
>
cournotbca
bcaqq NEWNEW =
->
-=+
3)(2
4)(3
21
16
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Observations• Iswhatwe’velookedatreallyasequentialgame?– Despitewesaidfirm1wasgoingtomovefirst,there’snoreasontoassumeshe’sreallygoingtodoso!
• Weneedacommitment• Inthisexample,sunkcostcouldhelpinbelievingfirm1willactuallyplayfirst
è Assumeforinstancefirm1wasgoingtoinvestalotofmoneyinbuildingaplanttosupportalargeproduction:thiswouldbeacrediblecommitment!
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Simultaneousvs.Sequential
• Therearesomekeyideasinvolvedhere1. Gamesbeingsimultaneousorsequentialis
notreallyabouttiming,itisaboutinformation
2. Sometimes,moreinformationcanhurt!3. Sometimes,moreoptionscanhurt!
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Firstmoveradvantage
• Advocatedbymany“economicsbooks”• Isbeingthefirstmoveralwaysgood?– Yes,sometimes:asintheStackelberg model– Notalways,asintheRock,Paper,Scissorsgame– Sometimesneitherbeingthefirstnorthesecondisgood,asinthe“Isplityouchoose”game
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TheNIMgame
• Wehavetwoplayers• Therearetwopilesofstones,AandB• Eachplayer,inturn,decidestodeletesomestonesfromwhateverpile
• Theplayerthatremainswiththelaststonewins
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TheNIMgame(2)
• Ifpilesareequalè secondmoveradvantage– Youwanttobeplayer2
• Ifpilesareunequalè firstmoveradvantage– Youwanttobeplayer1– Correcttactic:Youwanttomakepilesequal
• Youknowwhowillwinthegamefromtheinitialsetup
• Youcansolvethroughbackwardinduction
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Outline
1. Stackelberg duopolyandthefirstmover’sadvantage
2. Formaldefinitions3. Bargaininganddiscountedpayoffs
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PerfectInformationandpurestrategy
Agameofperfectinformation isoneinwhichateachnodeofthegametree,theplayerwhoseturnistomoveknowswhichnodesheisatandhowshegotthere
Apurestrategy forplayeri inagameofperfectinformationisacompleteplan ofactions:itspecifieswhichactioni willtakeateachofitsdecisionnodes
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Example
• Strategies– Player2:[l],[r]
– Player1:[U,u],[U,d][D,u],[D,d]
(1,0)
12
1
(0,2)
(2,4)
(3,1)U
D
l
rd
lookredundant!
u
• Note:– Inthisgameitappearsthatplayer2mayneverhavethepossibilitytoplayherstrategies
– Thisisalsotrueforplayer1! 24
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Backwardinductionsolution
• BackwardInduction– Startfromtheend• “d”à higherpayoff
– Summarizegame• “r”à higherpayoff
– Summarizegame• “D”à higherpayoff
(1,0)
12
1
(0,2)
(2,4)
(3,1)U
D
l
rd
u
• BI::{[D,d],r}
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Transformationtonormalform
2,4 0,2
3,1 0,2
1,0 1,0
1,0 1,0(1,0)
12
1
(0,2)
(2,4)
(3,1)U
D
l
rd
u
l r
Uu
Ud
Du
Dd
FromtheextensiveformTothenormalform
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BackwardinductionversusNE
2,4 0,2
3,1 0,2
1,0 1,0
1,0 1,0(1,0)
12
1
(0,2)
(2,4)
(3,1)U
D
l
rd
u
l r
Uu
Ud
Du
Dd
NashEquilibrium
{[D,d],r}{[D,u],r}
BackwardInduction
{[D,d],r}
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AMarketGame(1)
• Assumetherearetwoplayers– Anincumbentmonopolist(MicroSoft,MS)ofO.S.– Ayoungstart-upcompany(SU)withanewO.S.
• ThestrategiesavailabletoSUare:Enterthemarket(IN)orstayout(OUT)
• ThestrategiesavailabletoMSare:Lowerpricesanddomarketing(FIGHT)orstayput(NOTFIGHT)
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AMarketGame(2)
• Whatshouldyoudo?
• AnalyzethegamewithBI• AnalyzethenormalformequivalentandfindNE(0,3)
MS
(1,1)IN
OUT
F
NFSU
(-1,0)
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AMarketGame(3)
(0,3)
MS
(1,1)IN
OUT
F
NFSU
(-1,0)
-1,0 1,1
0,3 0,3
F NF
IN
OUT
NashEquilibrium
(IN,NF)(OUT,F)
BackwardInduction
(IN,NF)
• (OUT,FIGHT)isaNEbutreliesonanincrediblethreat– Introducesubgame perfectequilibrium
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Sub-games
• Asub-game isapartofthegamethatlookslikeagamewithinthetree.Itstartsfromasinglenodeandcomprisesallsuccessorsofthatnode
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sub-gameperfectequilibrium(SPE)
• ANashEquilibrium(s1*,s2*,…,sN*)isasub-gameperfectequilibriumifitinducesaNashEquilibriumineverysub-gameofthegame
• Example:– (IN,NF)isaSPE– (OUT,F)isnotaSPE• Incrediblethreat
(0,3)
MS
(1,1)IN
OUT
F
NFSU
(-1,0)
-1,0 1,1
0,3 0,3
F NF
IN
OUT 32
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Outline
1. Stackelberg duopolyandthefirstmover’sadvantage
2. Formaldefinitions3. Bargaininganddiscountedpayoffs
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Ultimatumgame• Twoplayers,player1isgoingtomakea“takeitorleaveit”offertoplayer2
• Player1isgivenapieworth$1andhastodecidehowtodivideit– (S,1-S),e.g.($0.75,$0.25)
• Player2hastwochoices:acceptordeclinetheoffer• Payoffs:– Ifplayer2accepts:Player1getsS,player2gets1-S– Ifplayer2declines:Player1andplayer2getnothing
• Itdoesn’tlooklikerealbargaining,but…let’splay34
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Analysiswithbackwardinduction
• Startwiththereceiveroftheoffer,choosingtoacceptorrefuse(1-S)– Assumingplayer2istryingtomaximizeherprofit,whatshouldshedo?
• So,whatshouldplayer1offer?
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Predictionvs reality• Isthereagoodmatchbetweenbackwardinduction
predictionandwhatweobserve?• Why?
• Reasonswhyplayer2mayreject:– Pride– Shemaybesensitivetohowherpayoffsrelatestoothers– Indignation– Player2maywantto“teach”alessontoPlayer1tooffermore
• Whatwereallyplayedisaone-shotgamebutifwehaveplayedmorethanonce,byrejectinganoffer,player2wouldalsoinduceplayer1toobtainnothing,whichmaybeanincentiveforplayer1tooffermoreinthenextroundofthegame
• Whyisthe50-50splitfocalhere?36
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Two-periodbargaininggame• Twoplayers,player1isgoingtomakea“takeitorleaveit”offertoplayer2
• Player1isgivenapieworth$1andhastodecidehowtodivideit:(S1,1-S1)
• Player2hastwochoices:acceptordeclinetheoffer– Ifplayer2accepts:Player1getsS1,player2gets1-S1– Ifplayer2declines:wefliptherolesandplayagain
• Thisisthesecondstageofthegame• Thesecondstageisexactlytheultimatumgame:player2choosesadivision(S2,1-S2)
• Player1canacceptorreject– Ifplayer1accepts,thedealisdone– Ifplayer1rejects,noneofthemgetsanything
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Discountfactor• Now,weaddoneimportantelement– Inthefirstround,thepieisworth$1– Ifweendupinthesecondround,thepieisworthless
• Example:– IfIgiveyou$1today,that’swhatyouget– IfIgiveyou$1in1month,weassumeit’sworthless,say
• Discountingfactor:– Fromtodayperspective,$1tomorrowisworth
€
δ <1
€
δ <138
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Gameanalysisidea• Itisclearthatthedecisiontoacceptorrejectpartlydependsonwhatyouthinktheothersideisgoingtodointhesecondround
èThisisbackwardinduction!– Byworkingbackwards,wecanseethatwhatyoushouldofferinthefirstroundshouldbejustenoughtomakesureit’saccepted,knowingthatthepersonwho’sreceivingtheofferinthefirstroundisgoingtothinkabouttheofferthey’regoingtomakeyouinthesecondround,andthey’regoingtothinkaboutwhetheryou’regoingtoacceptorreject
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Two-periodbargaininggameanalysis
• Let’sanalyzethegameformallywithbackwardinduction–Weignoreany“pride”effect
• Onestagegame(theultimatumgame)
Offerer’s split Receiver’ssplit
1-period 1 0
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Two-periodbargaininggameanalysis(2)
• Two-stagegame
Let’sbecareful:– Inthesecondroundofthetwo-periodgame,player2makestheofferabout
thewholepie– Weknowthatthisisgoingtobeanultimatumgame,soplayer2willkeepthe
wholepieandplayer1willaccept(byBI)– However,seenfromthefirstround,thepieinthesecondroundthatplayer2
couldget,isworthlessthan$1
Offerer’s split Receiver’ssplit
1-period 1 02-period
€
δ <1
€
1−δ
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Two-periodbargaininggamegraphically
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Two-periodbargaininggamegraphically(2)
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Three-periodbargaininggame
• Therulesarethesameasforthepreviousgames,butnowtherearetwopossibleflips– Period1:player1offersfirst– Period2:ifplayer2rejectedtheofferinperiod1,shegetstooffer
– Period3:ifplayer1rejectedtheofferinperiod2,hegetstoofferagain
• NOTE:thevalueofthepiekeepsshrinking– It’snotthepiethatreallyshrinks,it’sthatweassumedplayersarediscounting
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Three-periodbargaininggameanalysis
• Discounting:thevaluetoplayer1ofapieinroundthreeisdiscountedby
• Analysiswithbackwardinduction– Again,assume“nopride”–Westartfromroundthree,whichisourultimatumgameandweknowtherethatplayer1cangetthewholepie,sinceplayer2willaccepttheoffer
è Player1couldgetapieworth
€
δ⋅ δ = δ 2
€
δ 2
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Three-periodbargaininggameresult
• Three-periodgame
• NOTE:inthetable,wereportthesplitplayer1shouldofferinthefirstroundofthegame
• Inthefirstround,iftheofferisrejected,wegointoa2-periodgame,andweknowwhatthesplitisgoingtolooklike
Offerer’s split Receiver’ssplit
1-period 1 02-period3-period
€
δ <1
€
1−δ
€
δ 1−δ( )
€
1−δ 1−δ( )
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Three-periodbargaininggamegraphically
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Four-periods
• Whatabouta4-periodbargaininggame?
• NOTE:givepeoplejustenoughtodaysothey’llaccepttheoffer,andjustenoughtodayiswhatevertheygettomorrowdiscountedbydelta
• Youdon’tneedtogobackallthewayuptoperiod1
Offerer Receiver1-period 1 02-period3-period4-period ? ?
€
δ <1
€
1−δ
€
δ 1−δ( )
€
1−δ 1−δ( )
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Four-periodsresult
• Let’sclearoutthealgebra
Offerer Receiver1-period 1 02-period3-period4-period
€
δ
€
1−δ
€
δ −δ 2
€
1−δ +δ 2
€
δ −δ 2 +δ 3
€
1−δ +δ 2 −δ 349
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n-periods
• Geometricserieswithreason(-δ)• Forexample,player1’sshareforn=10:
50
S1(10) =1−δ +δ 2 −δ3 +δ 4 +...−δ 9 =
1− −δ( )10
1− (−δ)=1−δ10
1+δ
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Someobservations• Intheone-stagegame,there’sahugefirst-moveradvantage
• Inthetwo-stagegame,itsmoredifficult:itdependsonhowlargeisdelta.Ifitislarge,you’dpreferbeingthereceiver
• Inthethree-stagegameitlookslikeyou’dbebetteroffbymakingtheoffer,butagainit’snotveryeasy
• Whataboutthe10-stagegame?Itseemsthatthetwoplayersaregettingcloserintermsofpayoffs,andthattheinitialbargainingpowerhasdiminished
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Largenumberofperiods
• Let’slookattheasymptoticbehaviorofthisgame,whenthereisaninfinitenumberofstages
€
S1(∞) =
1−δ∞
1+δ=
11+δ
S2(∞) =1− S1
(∞) =δ +δ∞
1+δ=
δ1+δ
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Discountfactorclosetoone
• Now,let’simaginethattheoffersaremadeinrapidsuccession:thiswouldimplythatthediscountfactorwehintedatisalmostnegligible,whichboilsdowntoassumedeltatobeverycloseto1
• So,ifweassumerapidlyalternatingoffers,weendupwitha50-50split!€
S1(∞) =
11+δ
δ ≈1% → % 12
S2(∞) =
δ1+δ
δ ≈1% → % 12
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