GameTheory--

Lecture6

PatrickLoiseauEURECOMFall2016

1

Outline

1. Stackelberg duopolyandthefirstmover’sadvantage

2. Formaldefinitions3. Bargaininganddiscountedpayoffs

2

Outline

1. Stackelberg duopolyandthefirstmover’sadvantage

2. Formaldefinitions3. Bargaininganddiscountedpayoffs

3

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

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=

Graphically

6

0 q1

q2

bca

2-

bca -

NEMonopoly

Perfectcompetition

BR2

BR1

bcaqCournot 3

-=

Stackelberg Model• Assumenowthatonefirmgetstomovefirstandtheothermovesafter– Thatisonefirmgetstosetthequantityfirst

• Isitanadvantagetomovefirst?– Oritisbettertowaitandseewhattheotherfirmisdoingandthenreact?

• Wearegoingtousebackwardinductiontocomputethequantities– Wecannotdrawtreesherebecauseofthecontinuumofpossibleactions

7

Intuition

• Suppose1movesfirst• 2respondsbyBR!(bydef)• Whatquantityshouldfirm1produce,knowingthatfirm2willrespondusingtheBR?– constrainedoptimizationproblem

80 q1

q2

BR2

q’1q’’1

q’2

q’’2

Intuition(2)• Shouldfirm1producemoreorlessthantheCournotquantity?– Productsarestrategicsubstitutes:themorefirm1produces,thelessfirm2willproduceandvice-versa

– Firm1producingmoreè firm1ishappy• Whathappenstofirm1’sprofits?– Theygoup,otherwisefirm1wouldn’thavesethigherproductionquantities

• Whathappenstofirm2’sprofits?– Theanswerisnotimmediate

• Whathappenedtothetotaloutputinthemarket?– Evenheretheanswerisnotimmediate

9

Intuition(3)

• Whathappenedtothetotaloutputinthemarket?– Consumerswouldlikethetotaloutputtogoup,forthatwouldmeanthatpriceswouldgodown!

– Indeed,itgoesdown:seetheBRcurve

10

0 q1

q2

BR2

q’’1q’1

q’’2q’2

Theincrementfromq’1toq’’1islargerthanthedecrementfromq’2toq’’2

Intuition(4)

• Whathappenstofirm2’sprofits?– q1wentup,q2wentdown– q1+q2wentupè priceswentdown– Firm2’scostsarethesame

èFirm2’sprofitwentdown

• Wehaveseenthatfirm1’sprofitgoesup

èConclusion:Firstmoverisanasset(here!) 11

Stackelberg Modelcomputations

• Letusnowcomputethequantities.Wehave

• WeapplytheBackwardInductionprinciple– First,solvethemaximizationproblemforfirm2,takingq1asgiven

– Then,focusonfirm1

€

p = a − b(q1 + q2)profit i = pqi − cqi

12

Stackelberg Modelcomputations(2)

• Firm2’soptimizationproblem(forfixedq1)

• Wenowcantakethisquantityandplugitinthemaximizationproblemforfirm1

( )[ ]

22

max

12

2

22212

qbcaq

q

cqqbqbqaq

--

=Þ¶¶

---

13

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)

* +

,

- .

14

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

Stackelberg quantities

• Allthismathtoverifyourinitialintuition!

CournotNEW

CournotNEW

qqqq

22

11

<

>

cournotbca

bcaqq NEWNEW =

->

-=+

3)(2

4)(3

21

16

Observations• Iswhatwe’velookedatreallyasequentialgame?– Despitewesaidfirm1wasgoingtomovefirst,there’snoreasontoassumeshe’sreallygoingtodoso!

• Weneedacommitment• Inthisexample,sunkcostcouldhelpinbelievingfirm1willactuallyplayfirst

è Assumeforinstancefirm1wasgoingtoinvestalotofmoneyinbuildingaplanttosupportalargeproduction:thiswouldbeacrediblecommitment!

17

Simultaneousvs.Sequential

• Therearesomekeyideasinvolvedhere1. Gamesbeingsimultaneousorsequentialis

notreallyabouttiming,itisaboutinformation

2. Sometimes,moreinformationcanhurt!3. Sometimes,moreoptionscanhurt!

18

Firstmoveradvantage

• Advocatedbymany“economicsbooks”• Isbeingthefirstmoveralwaysgood?– Yes,sometimes:asintheStackelberg model– Notalways,asintheRock,Paper,Scissorsgame– Sometimesneitherbeingthefirstnorthesecondisgood,asinthe“Isplityouchoose”game

19

TheNIMgame

• Wehavetwoplayers• Therearetwopilesofstones,AandB• Eachplayer,inturn,decidestodeletesomestonesfromwhateverpile

• Theplayerthatremainswiththelaststonewins

20

TheNIMgame(2)

• Ifpilesareequalè secondmoveradvantage– Youwanttobeplayer2

• Ifpilesareunequalè firstmoveradvantage– Youwanttobeplayer1– Correcttactic:Youwanttomakepilesequal

• Youknowwhowillwinthegamefromtheinitialsetup

• Youcansolvethroughbackwardinduction

21

Outline

1. Stackelberg duopolyandthefirstmover’sadvantage

2. Formaldefinitions3. Bargaininganddiscountedpayoffs

22

PerfectInformationandpurestrategy

Agameofperfectinformation isoneinwhichateachnodeofthegametree,theplayerwhoseturnistomoveknowswhichnodesheisatandhowshegotthere

Apurestrategy forplayeri inagameofperfectinformationisacompleteplan ofactions:itspecifieswhichactioni willtakeateachofitsdecisionnodes

23

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

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}

25

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

26

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}

27

AMarketGame(1)

• Assumetherearetwoplayers– Anincumbentmonopolist(MicroSoft,MS)ofO.S.– Ayoungstart-upcompany(SU)withanewO.S.

• ThestrategiesavailabletoSUare:Enterthemarket(IN)orstayout(OUT)

• ThestrategiesavailabletoMSare:Lowerpricesanddomarketing(FIGHT)orstayput(NOTFIGHT)

28

AMarketGame(2)

• Whatshouldyoudo?

• AnalyzethegamewithBI• AnalyzethenormalformequivalentandfindNE(0,3)

MS

(1,1)IN

OUT

F

NFSU

(-1,0)

29

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

30

Sub-games

• Asub-game isapartofthegamethatlookslikeagamewithinthetree.Itstartsfromasinglenodeandcomprisesallsuccessorsofthatnode

31

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

Outline

1. Stackelberg duopolyandthefirstmover’sadvantage

2. Formaldefinitions3. Bargaininganddiscountedpayoffs

33

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

Analysiswithbackwardinduction

• Startwiththereceiveroftheoffer,choosingtoacceptorrefuse(1-S)– Assumingplayer2istryingtomaximizeherprofit,whatshouldshedo?

• So,whatshouldplayer1offer?

35

Predictionvs reality• Isthereagoodmatchbetweenbackwardinduction

predictionandwhatweobserve?• Why?

• Reasonswhyplayer2mayreject:– Pride– Shemaybesensitivetohowherpayoffsrelatestoothers– Indignation– Player2maywantto“teach”alessontoPlayer1tooffermore

• Whatwereallyplayedisaone-shotgamebutifwehaveplayedmorethanonce,byrejectinganoffer,player2wouldalsoinduceplayer1toobtainnothing,whichmaybeanincentiveforplayer1tooffermoreinthenextroundofthegame

• Whyisthe50-50splitfocalhere?36

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

37

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

Gameanalysisidea• Itisclearthatthedecisiontoacceptorrejectpartlydependsonwhatyouthinktheothersideisgoingtodointhesecondround

èThisisbackwardinduction!– Byworkingbackwards,wecanseethatwhatyoushouldofferinthefirstroundshouldbejustenoughtomakesureit’saccepted,knowingthatthepersonwho’sreceivingtheofferinthefirstroundisgoingtothinkabouttheofferthey’regoingtomakeyouinthesecondround,andthey’regoingtothinkaboutwhetheryou’regoingtoacceptorreject

39

Two-periodbargaininggameanalysis

• Let’sanalyzethegameformallywithbackwardinduction–Weignoreany“pride”effect

• Onestagegame(theultimatumgame)

Offerer’s split Receiver’ssplit

1-period 1 0

40

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

41

Two-periodbargaininggamegraphically

42

Two-periodbargaininggamegraphically(2)

43

Three-periodbargaininggame

• Therulesarethesameasforthepreviousgames,butnowtherearetwopossibleflips– Period1:player1offersfirst– Period2:ifplayer2rejectedtheofferinperiod1,shegetstooffer

– Period3:ifplayer1rejectedtheofferinperiod2,hegetstoofferagain

• NOTE:thevalueofthepiekeepsshrinking– It’snotthepiethatreallyshrinks,it’sthatweassumedplayersarediscounting

44

Three-periodbargaininggameanalysis

• Discounting:thevaluetoplayer1ofapieinroundthreeisdiscountedby

• Analysiswithbackwardinduction– Again,assume“nopride”–Westartfromroundthree,whichisourultimatumgameandweknowtherethatplayer1cangetthewholepie,sinceplayer2willaccepttheoffer

è Player1couldgetapieworth

€

δ⋅ δ = δ 2

€

δ 2

45

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( )

46

Three-periodbargaininggamegraphically

47

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( )

48

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

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

Someobservations• Intheone-stagegame,there’sahugefirst-moveradvantage

• Inthetwo-stagegame,itsmoredifficult:itdependsonhowlargeisdelta.Ifitislarge,you’dpreferbeingthereceiver

• Inthethree-stagegameitlookslikeyou’dbebetteroffbymakingtheoffer,butagainit’snotveryeasy

• Whataboutthe10-stagegame?Itseemsthatthetwoplayersaregettingcloserintermsofpayoffs,andthattheinitialbargainingpowerhasdiminished

51

Largenumberofperiods

• Let’slookattheasymptoticbehaviorofthisgame,whenthereisaninfinitenumberofstages

€

S1(∞) =

1−δ∞

1+δ=

11+δ

S2(∞) =1− S1

(∞) =δ +δ∞

1+δ=

δ1+δ

52

Discountfactorclosetoone

• Now,let’simaginethattheoffersaremadeinrapidsuccession:thiswouldimplythatthediscountfactorwehintedatisalmostnegligible,whichboilsdowntoassumedeltatobeverycloseto1

• So,ifweassumerapidlyalternatingoffers,weendupwitha50-50split!€

S1(∞) =

11+δ

δ ≈1% → % 12

S2(∞) =

δ1+δ

δ ≈1% → % 12

53