GameTheory
-Lecture6
PatrickLoiseau
EURECOM
Fall2016
1
Outline
1. Stackelberg duopolyandthefirstmover’s
advantage
2. Formaldefinitions
3. Bargaininganddiscountedpayoffs
2
Outline
1. Stackelberg duopolyandthefirstmover’s
advantage
2. Formaldefinitions
3. Bargaininganddiscountedpayoffs
3
Cournot Competitionreminder
• Theplayers:2Firms,e.g.CokeandPepsi
• Strategies:quantitiesplayersproduceofidentical
products:qi,q-i
– Productsareperfectsubstitutes
• Thepayoffs
– Constantmarginalcostofproductionc
– Marketclearingprice:p=a– b(q1 +q2)
– firmsaimtomaximizeprofit
p
a
0
Demand
Slope:-b curve
q1 +q2
u1(q1,q2)=p*q1 – c*q1
4
Nashequilibrium
• u1(q1,q2)=a*q1 – b*q21 – b*q1 q2 – c*q1
• FOC,SOCgivebestresponses: ìqˆ1 = BR1 (q2 ) = a - c - q2
ïï
2b
2
í
ïqˆ = BR (q ) = a - c - q1
2
1
ïî 2
2b
2
• NEiswhentheycross:
BR1 (q2 ) = BR2 (q1 ) Þ q1* = q2*
a - c q2 a - c q1
- =
2b
2
2b
2
a-c
*
*
Þ q1 = q2 =
3b
à Cournot quantity
5
Graphically
q2
a-c
b
BR1
Monopoly
NE
qCournot =
a-c
3b
BR2
0
a-c
2b
Perfect
competition
q1
6
Stackelberg Model
• Assumenowthatonefirmgetstomovefirstand
theothermovesafter
– Thatisonefirmgetstosetthequantityfirst
• Isitanadvantagetomovefirst?
– Oritisbettertowaitandseewhattheotherfirmis
doingandthenreact?
• Wearegoingtousebackwardinductionto
computethequantities
– Wecannotdrawtreesherebecauseofthecontinuum
ofpossibleactions
7
Intuition
• Suppose1movesfirst
• 2respondsbyBR!(bydef)
• Whatquantityshouldfirm1
produce,knowingthatfirm2
willrespondusingtheBR?
q2
– constrainedoptimizationproblem
q’’2
BR2
q’2
0
q’’1
q’1
q1
8
Intuition(2)
• Shouldfirm1producemoreorlessthantheCournot
quantity?
– Productsarestrategicsubstitutes:themorefirm1
produces,thelessfirm2willproduceandvice-versa
– Firm1producingmoreè firm1ishappy
• Whathappenstofirm1’sprofits?
– Theygoup,otherwisefirm1wouldn’thavesethigher
productionquantities
• Whathappenstofirm2’sprofits?
– Theanswerisnotimmediate
• Whathappenedtothetotaloutputinthemarket?
– Evenheretheanswerisnotimmediate
9
Intuition(3)
• Whathappenedtothe
totaloutputinthe
market?
– Consumerswouldlike q2
thetotaloutputtogo
up,forthatwouldmean
thatpriceswouldgo
down!
– Indeed,itgoesdown: q’2
q’’2
seetheBRcurve
0
Theincrementfromq’1toq’’1
islargerthanthedecrementfrom
q’2toq’’2
BR2
q’1
q1
q’’1
10
Intuition(4)
• Whathappenstofirm2’sprofits?
– q1wentup,q2wentdown
– q1+q2wentupè priceswentdown
– Firm2’scostsarethesame
èFirm2’sprofitwentdown
• Wehaveseenthatfirm1’sprofitgoesup
èConclusion:Firstmoverisanasset(here!)
11
Stackelberg Modelcomputations
• Letusnowcomputethequantities.Wehave
p = a − b(q1 + q2 )
profit i = pqi − cqi
• WeapplytheBackwardInductionprinciple
– First,solvethemaximizationproblemforfirm2,
takingq1asgiven
– Then,focusonfirm1
12
Stackelberg Modelcomputations(2)
• Firm2’soptimizationproblem(forfixedq1)
max[(a - bq1 - bq2 )q2 - cq2 ]
q2
¶
a - c q1
Þ q2 =
¶q2
2b
2
• Wenowcantakethisquantityandplugitin
themaximizationproblemforfirm1
13
Stackelberg Modelcomputations(3)
• Firm1’soptimizationproblem:
[
]
max ( a − bq1 − bq2 )q1 − cq1 =
q1
)#
# a − c q1 && ,
max+% a − bq1 − b%
− (( − c .q1 =
q1 *$
$ 2b
2 '' 2,
)a − c
) a − c bq1 ,
q1
max+
−
q1 = max+
q1 − b .
.
q1 * 2
q1 * 2
2 214
Stackelberg Modelcomputations(4)
• WederiveF.O.C.andS.O.C.
¶
a-c
=0Þ
- bq1 = 0
¶q1
2
¶2
= -b < 0
2
¶q1
• Thisgivesus
a-c
q1 =
2b
a-c 1 a-c a-c
q2 =
=
2b 2 2b
4b
15
Stackelberg quantities
• Allthismathtoverifyourinitialintuition!
NEW
1
>q
NEW
2
<q
q
q
NEW
1
q
Cournot
1
+q
Cournot
2
NEW
2
3(a - c) 2(a - c)
=
>
= cournot
4b
3b
16
Observations
• Iswhatwe’velookedatreallyasequentialgame?
– Despitewesaidfirm1wasgoingtomovefirst,there’sno
reasontoassumeshe’sreallygoingtodoso!
• Weneedacommitment
• Inthisexample,sunkcostcouldhelpinbelievingfirm1
willactuallyplayfirst
è Assumeforinstancefirm1wasgoingtoinvestalotof
moneyinbuildingaplanttosupportalarge
production:thiswouldbeacrediblecommitment!
17
Simultaneousvs.Sequential
• Therearesomekeyideasinvolvedhere
1. Gamesbeingsimultaneousorsequentialis
notreallyabouttiming,itisabout
information
2. Sometimes,moreinformationcanhurt!
3. Sometimes,moreoptionscanhurt!
18
Firstmoveradvantage
• Advocatedbymany“economicsbooks”
• Isbeingthefirstmoveralwaysgood?
– Yes,sometimes:asintheStackelberg model
– Notalways,asintheRock,Paper,Scissorsgame
– Sometimesneitherbeingthefirstnorthesecond
isgood,asinthe“Isplityouchoose”game
19
TheNIMgame
• Wehavetwoplayers
• Therearetwopilesofstones,AandB
• Eachplayer,inturn,decidestodeletesome
stonesfromwhateverpile
• Theplayerthatremainswiththelaststone
wins
20
TheNIMgame(2)
• Ifpilesareequalè secondmoveradvantage
– Youwanttobeplayer2
• Ifpilesareunequalè firstmoveradvantage
– Youwanttobeplayer1
– Correcttactic:Youwanttomakepilesequal
• Youknowwhowillwinthegamefromtheinitial
setup
• Youcansolvethroughbackwardinduction
21
Outline
1. Stackelberg duopolyandthefirstmover’s
advantage
2. Formaldefinitions
3. Bargaininganddiscountedpayoffs
22
PerfectInformationandpurestrategy
Agameofperfectinformation isoneinwhichat
eachnodeofthegametree,theplayerwhose
turnistomoveknowswhichnodesheisatand
howshegotthere
Apurestrategy forplayeri inagameofperfect
informationisacompleteplan ofactions:it
specifieswhichactioni willtakeateachofits
decisionnodes
23
Example
• Strategies
1
2
1
U
D
l
r
(1,0)
u
(2,4)
d
(3,1)
(0,2)
– Player2:
[l],[r]
– Player1:
[U,u],[U,d]
[D,u],[D,d]
lookredundant!
• Note:
– Inthisgameitappearsthatplayer2mayneverhavethe
possibilitytoplayherstrategies
– Thisisalsotrueforplayer1!
24
Backwardinductionsolution
• BackwardInduction
1
2
1
U
D
l
r
(1,0)
u
(2,4)
d
(3,1)
(0,2)
– Startfromtheend
• “d”à higherpayoff
– Summarizegame
• “r”à higherpayoff
– Summarizegame
• “D”à higherpayoff
• BI::{[D,d],r}
25
Transformationtonormalform
1
2
1
U
D
l
r
(1,0)
u
d
(0,2)
(2,4)
(3,1)
l
r
Uu
2,4
0,2
Ud
3,1
0,2
Du
1,0
1,0
Dd
1,0
1,0
Fromtheextensiveform
Tothenormalform
26
BackwardinductionversusNE
1
2
1
U
D
l
r
u
d
(2,4)
(3,1)
(0,2)
(1,0)
l
r
Uu
2,4
0,2
Ud
3,1
0,2
Du
1,0
1,0
Dd
1,0
1,0
BackwardInduction
NashEquilibrium
{[D,d],r}
{[D,d],r}
{[D,u],r}
27
AMarketGame(1)
• Assumetherearetwoplayers
– Anincumbentmonopolist(MicroSoft,MS)ofO.S.
– Ayoungstart-upcompany(SU)withanewO.S.
• ThestrategiesavailabletoSUare:
Enterthemarket(IN)orstayout(OUT)
• ThestrategiesavailabletoMSare:
Lowerpricesanddomarketing(FIGHT)orstay
put(NOTFIGHT)
28
AMarketGame(2)
• Whatshouldyoudo?
MS F
SU IN
OUT
NF
(0,3)
(-1,0)
(1,1)
• AnalyzethegamewithBI
• Analyzethenormalform
equivalentandfindNE
29
AMarketGame(3)
MS F
SU IN
NF
OUT
F
NF
IN
-1,0
1,1
OUT
0,3
0,3
(-1,0)
(1,1)
(0,3)
BackwardInduction
NashEquilibrium
(IN,NF)
(IN,NF)
(OUT,F)
• (OUT,FIGHT)isaNEbutreliesonanincrediblethreat
– Introducesubgame perfectequilibrium
30
Sub-games
• Asub-game isapartofthegamethatlookslikea
gamewithinthetree.Itstartsfromasinglenode
andcomprisesallsuccessorsofthatnode
31
sub-gameperfectequilibrium(SPE)
• ANashEquilibrium(s1*,s2*,…,sN*)isasubgameperfectequilibriumifitinducesaNash
Equilibriumineverysub-gameofthegame
• Example:
– (IN,NF)isaSPE
– (OUT,F)isnotaSPE
MS F
SU IN
OUT
• Incrediblethreat
NF
(-1,0)
(1,1)
(0,3)
F
NF
IN
-1,0
1,1
OUT
0,3
0,3
32
Outline
1. Stackelberg duopolyandthefirstmover’s
advantage
2. Formaldefinitions
3. Bargaininganddiscountedpayoffs
33
Ultimatumgame
• Twoplayers,player1isgoingtomakea“takeitor
leaveit”offertoplayer2
• Player1isgivenapieworth$1andhastodecidehow
todivideit
– (S,1-S),e.g.($0.75,$0.25)
• Player2hastwochoices:acceptordeclinetheoffer
• Payoffs:
– Ifplayer2accepts:Player1getsS,player2gets1-S
– Ifplayer2declines:Player1andplayer2getnothing
• Itdoesn’tlooklikerealbargaining,but…let’splay
34
Analysiswithbackwardinduction
• Startwiththereceiveroftheoffer,choosing
toacceptorrefuse(1-S)
– Assumingplayer2istryingtomaximizeherprofit,
whatshouldshedo?
• So,whatshouldplayer1offer?
35
Predictionvs reality
• Isthereagoodmatchbetweenbackwardinduction
predictionandwhatweobserve?
• Why?
• Reasonswhyplayer2mayreject:
–
–
–
–
Pride
Shemaybesensitivetohowherpayoffsrelatestoothers
Indignation
Player2maywantto“teach”alessontoPlayer1tooffermore
• Whatwereallyplayedisaone-shotgamebutifwehaveplayedmore
thanonce,byrejectinganoffer,player2wouldalsoinduceplayer1to
obtainnothing,whichmaybeanincentiveforplayer1tooffermore
inthenextroundofthegame
• Whyisthe50-50splitfocalhere?
36
Two-periodbargaininggame
• Twoplayers,player1isgoingtomakea“takeitor
leaveit”offertoplayer2
• Player1isgivenapieworth$1andhastodecidehow
todivideit:(S1,1-S1)
• Player2hastwochoices:acceptordeclinetheoffer
– Ifplayer2accepts:Player1getsS1,player2gets1-S1
– Ifplayer2declines:wefliptherolesandplayagain
• Thisisthesecondstageofthegame
• Thesecondstageisexactlytheultimatumgame:player
2choosesadivision(S2,1-S2)
• Player1canacceptorreject
– Ifplayer1accepts,thedealisdone
– Ifplayer1rejects,noneofthemgetsanything
37
Discountfactor
• Now,weaddoneimportantelement
– Inthefirstround,thepieisworth$1
– Ifweendupinthesecondround,thepieisworthless
• Example:
– IfIgiveyou$1today,that’swhatyouget
– IfIgiveyou$1in1month,weassumeit’sworthless,say
δ <1
• Discountingfactor:
– Fromtodayperspective,$1tomorrowisworth
δ <1
38
Gameanalysisidea
• Itisclearthatthedecisiontoacceptorreject
partlydependsonwhatyouthinktheotherside
isgoingtodointhesecondround
èThisisbackwardinduction!
– Byworkingbackwards,wecanseethatwhatyou
shouldofferinthefirstroundshouldbejustenough
tomakesureit’saccepted,knowingthattheperson
who’sreceivingtheofferinthefirstroundisgoingto
thinkabouttheofferthey’regoingtomakeyouinthe
secondround,andthey’regoingtothinkabout
whetheryou’regoingtoacceptorreject
39
Two-periodbargaininggameanalysis
• Let’sanalyzethegameformallywith
backwardinduction
– Weignoreany“pride”effect
• Onestagegame(theultimatumgame)
1-period
Offerer’s split
Receiver’ssplit
1
0
40
Two-periodbargaininggameanalysis
(2)
• Two-stagegame
1-period
2-period
Offerer’s split
Receiver’ssplit
1
0
1−δ
δ <1
Let’sbecareful:
– Inthesecondroundofthetwo-periodgame,player2makestheofferabout
thewholepie
– Weknowthatthisisgoingtobeanultimatumgame,soplayer2willkeepthe
wholepieandplayer1willaccept(byBI)
– However,seenfromthefirstround,thepieinthesecondroundthatplayer2
41
couldget,isworthlessthan$1
€
€
Two-periodbargaininggame
graphically
42
Two-periodbargaininggame
graphically(2)
43
Three-periodbargaininggame
• Therulesarethesameasforthepreviousgames,
butnowtherearetwopossibleflips
– Period1:player1offersfirst
– Period2:ifplayer2rejectedtheofferinperiod1,she
getstooffer
– Period3:ifplayer1rejectedtheofferinperiod2,he
getstoofferagain
• NOTE:thevalueofthepiekeepsshrinking
– It’snotthepiethatreallyshrinks,it’sthatweassumed
playersarediscounting
44
Three-periodbargaininggameanalysis
• Discounting:thevaluetoplayer1ofapiein
roundthreeisdiscountedby δ ⋅ δ = δ 2
• Analysiswithbackwardinduction
– Again,assume“nopride”
– Westartfromroundthree,whichisour
€
ultimatumgameandweknowtherethatplayer1
cangetthewholepie,sinceplayer2willaccept
theoffer
è Player1couldgetapieworth δ 2
45
Three-periodbargaininggameresult
• Three-periodgame
1-period
2-period
3-period
•
•
Offerer’s split
Receiver’ssplit
1
0
1−δ
δ <1
1 − δ (1 − δ )
δ (1 − δ )
NOTE:inthetable,wereportthesplitplayer1shouldofferinthefirst
€
€
roundofthegame
Inthefirstround,iftheofferisrejected,wegointoa2-periodgame,and
weknowwhatthesplitisgoingtolooklike
€
€
46
Three-periodbargaininggame
graphically
47
Four-periods
• Whatabouta4-periodbargaininggame?
Offerer
1
1-period
1−δ
2-period
3-period 1 − δ (1 − δ )
4-period
?
•
•
€
€
Receiver
0
δ <1
δ (1 − δ )
?
NOTE:givepeoplejustenoughtodaysothey’llaccepttheoffer,andjust
enoughtodayiswhatevertheygettomorrowdiscountedbydelta
€
€
48
Youdon’tneedtogobackallthewayuptoperiod1
Four-periodsresult
• Let’sclearoutthealgebra
1-period
2-period
3-period
4-period
€
Offerer
1
Receiver
0
1−δ
δ
1−δ +δ2
δ −δ2
1−δ +δ2 −δ3
€
δ −δ2 +δ3
49
n-periods
• Geometricserieswithreason(-δ)
• Forexample,player1’sshareforn=10:
10
10
1−
−
δ
1− δ
(
)
(10)
2
3
4
9
S1 = 1− δ + δ − δ + δ +... − δ =
=
1− (−δ )
1+ δ
50
Someobservations
• Intheone-stagegame,there’sahugefirst-mover
advantage
• Inthetwo-stagegame,itsmoredifficult:itdependson
howlargeisdelta.Ifitislarge,you’dpreferbeingthe
receiver
• Inthethree-stagegameitlookslikeyou’dbebetteroff
bymakingtheoffer,butagainit’snotveryeasy
• Whataboutthe10-stagegame?Itseemsthatthetwo
playersaregettingcloserintermsofpayoffs,andthat
theinitialbargainingpowerhasdiminished
51
Largenumberofperiods
• Let’slookattheasymptoticbehaviorofthis
game,whenthereisaninfinitenumberof
stages
∞
(∞)
1
S
S2(∞)
1−δ
1
=
=
1+ δ
1+ δ
∞
δ
+
δ
δ
(∞)
= 1 − S1 =
=
1+ δ
1+ δ
52
Discountfactorclosetoone
• Now,let’simaginethattheoffersaremadein
rapidsuccession:thiswouldimplythatthe
discountfactorwehintedatisalmostnegligible,
whichboilsdowntoassumedeltatobevery
closeto1
1
1
δ ≈1
(∞)
S1 =
%%→
1+ δ
2
δ
1
δ ≈1
(∞)
S2 =
%%→
1+ δ
2
• So,ifweassumerapidlyalternatingoffers,we
endupwitha50-50split!
53