Game Theory -- Lecture 2 - EURECOM

GameTheory--

Lecture2

PatrickLoiseauEURECOMFall2016

1

Lecture1recap

• Definedgamesinnormalform• Defineddominancenotion– Iterativedeletion– Doesnotalwaysgiveasolution

• DefinedbestresponseandNashequilibrium– ComputedNashequilibriuminsomeexamples

à AresomeNashequilibria betterthanothers?à CanwealwaysfindaNashequilibrium?

2

Outline

1. CoordinationgamesandParetooptimality2. Gameswithcontinuousactionsets– Equilibriumcomputationandexistencetheorem– Example:Cournot duopoly

3

Outline


4

TheInvestmentGame• Theplayers:you• Thestrategies:eachofyouchoosesbetweeninvestingnothinginaclassproject($0)orinvesting($10)

• Payoffs:– Ifyoudon’tinvestyourpayoffis$0– Ifyouinvestyoumakeanetprofitof$5(grossprofit=$15;investment$10) ifmorethan90%oftheclasschoosestoinvest.Otherwise,youlose$10

• Chooseyouraction(nocommunication!)

5

Nashequilibrium

• WhataretheNashequilibria?

• Remark:tofindNashequilibria,weuseda“guessandcheckmethod”– Checkingiseasy,guessingcanbehard

6

TheInvestmentGameagain• Recallthat:– Players:you– Strategies:invest$0orinvest$10– Payoffs:

• Ifnoinvestà $0$5netprofitif≥90%invest

• Ifinvest$10à-$10netprofitif<90%invest

• Let’splayagain!(nocommunication)

• WeareheadingtowardanequilibriumèTherearecertaincasesinwhichplayingconvergesinanaturalsensetoanequilibrium 7

Paretodomination

• Isoneequilibriumbetterthantheother?

• Intheinvestmentgame?8

Definition: ParetodominationAstrategyprofilesParetodominatesstrategyprofiles’iif foralli,ui(s)≥ui(s’)andthereexistsjsuchthatuj(s)>uj(s’);i.e.,allplayershaveatleastashighpayoffsandatleastoneplayerhasstrictlyhigherpayoff.

ConvergencetoequilibriumintheInvestmentGame

• WhydidweconvergetothewrongNE?• Rememberwhenwestartedplaying– Weweremoreorless50%investing

• Thestartingpointwasalreadybadforthepeoplewhoinvestedforthemtoloseconfidence

• Thenwejusttumbleddown

• Whatwouldhavehappenedifwestartedwith95%oftheclassinvesting?

9

Coordinationgame• Thisisacoordinationgame

– We’dlikeeveryonetocoordinatetheiractionsandinvest• Manyotherexamplesofcoordinationgames

– PartyinaVilla– On-lineWebSites– Establishmentoftechnologicalmonopolies(Microsoft,HDTV)– Bankruns

• Unlikeinprisoner’sdilemma,communicationhelps incoordinationgamesà scopeforleadership– Inprisoner’sdilemma,atrustedthirdparty(TTP)wouldneedto

imposeplayerstoadoptastrictlydominatedstrategy– Incoordinationgames,aTTPjustleadsthecrowdtowardsa

betterNEpoint(thereisnodominatedstrategy)

10

Battleofthesexes

• FindtheNE

• IsthereaNEbetterthantheother(s)?

2,1 0,00,0 1,2

Opera

Soccer

Opera

Player1

Player2Soccer

11

CoordinationGames• Purecoordinationgames:thereisnoconflictwhetheroneNEisbetterthantheother– E.g.:intheinvestmentgame,weallagreedthattheNEwitheveryoneinvestingwasa“better”NE

• Generalcoordinationgames:thereisasourceofconflictasplayerswouldagreetocoordinate,butoneNEis“better”foraplayerandnotfortheother– E.g.:BattleoftheSexes

è Communicationmightfailinthiscase

12

Paretooptimality

• Battleofthesexes?

13

Definition: ParetooptimalityAstrategyprofilesisParetooptimaliftheredoesnotexistastrategyprofiles’thatParetodominatess.

Outline


14

Thepartnershipgame(seeexercisesheet2)

• Twopartnerschooseeffortsi inSi=[0,4]• Sharerevenueandhavequadraticcosts

u1(s1 ,s2)=½[4(s1 +s2 +bs1 s2)]- s12

u2(s1 ,s2)=½[4(s1 +s2 +bs1 s2)]- s22

• Bestresponses:ŝ1 =1+bs2 =BR1(s2)ŝ2 =1+bs1 =BR2(s1)

15

Findingthebestresponse(withtwicecontinuouslydifferentiableutilities)∂u1(s1, s2 )

∂s1= 0

∂2u1(s1, s2 )∂2s1

≤ 0

• Firstordercondition(FOC)

• Secondordercondition(SOC)

• Remark:theSOCisautomaticallysatisfiedifui(si,s-i)isconcaveinsi foralls-i (verystandardassumption)

• Remark2:becarefulwiththeborders!– Exampleu1(s1,s2)=10-(s1+s2)2– S1=[0,4],whatistheBRtos2=2?– SolvingtheFOC,whatdoweget?

– WhentheFOCsolutionisoutsideSi,theBRisattheborder16

Nashequilibriumgraphically

• NEisfixedpointof(s1,s2)à (BR(s2),BR(s1)) 17

0

5

4

3

2

1

54321 s1

s2

BR1(s2)

BR2(s1)

Bestresponsecorrespondence

• Definition:ŝi isaBRtos-i ifŝi solvesmax ui(si ,s-i)• TheBRtos-i maynotbeunique!• BR(s-i):setofsi thatsolvemax ui(si ,s-i)• Thedefinitioncanbewritten:ŝi isaBRtos-i if

• Bestresponsecorrespondenceofi:s-i à BRi(s-i)• (Correspondence=set-valuedfunction)

18

si ∈ BRi (s−i ) = argmaxsi

ui (si, s−i )

Nashequilibriumasafixedpoint

• Game• Let’sdefine(setofstrategyprofiles)andthecorrespondence

• Foragivens,B(s)isthesetofstrategyprofiless’suchthatsi’isaBRtos-i foralli.

• Astrategyprofiles* isaNasheq.iif(justare-writingofthedefinition)

19

N, Si( )i∈N , ui( )i∈N( )S = ×i∈N Si

B : S→ S s B(s) = ×i∈N BRi (s−i )

s* ∈ B(s*)

Kakutani’s fixedpointtheorem

20

Theorem: Kakutani’s fixedpointtheoremLetX beacompactconvexsubsetofRn andlet

beaset-valuedfunctionforwhich:• forall,thesetisnonemptyconvex;• thegraphoffisclosed.Thenthereexistssuchthat x* ∈ f (x*)x* ∈ X

x ∈ X f (x)f : X→ X

Closedgraph(upperhemicontinuity)

• Definition:fhasclosedgraphifforallsequences(xn)and(yn)suchthatyn isinf(xn)foralln,xnàx andynày,yisinf(x)

• Alternativedefinition:fhasclosedgraphif forallxwehavethefollowingproperty:foranyopenneighborhoodVoff(x),thereexistsaneighborhoodUofxsuchthatforallxinU,f(x)isasubsetofV.

• Examples:

21

Existenceof(purestrategy)Nashequilibrium

• Remark:theconcaveassumptioncanberelaxed22

Theorem: ExistenceofpurestrategyNESupposethatthe gamesatisfies:• Theactionsetofeachplayerisanonempty

compactconvexsubsetofRn

• Theutilityofeachplayeriscontinuousin(on)andconcavein(on)

Then,thereexistsa(purestrategy)Nashequilibrium.

N, Si( )i∈N , ui( )i∈N( )Si

ui ssi SiS

Proof• DefineBasbefore.BsatisfiestheassumptionsofKakutani’s fixedpointtheorem

• ThereforeBhasafixedpointwhichbydefinitionisaNashequilibrium!

• Now,weneedtoactuallyverifythatBsatisfiestheassumptionsofKakutani’s fixedpointtheorem!

23

Example:thepartnershipgame• N={1,2}• S=[0,4]x[0,4]compactconvex• Utilitiesarecontinuousandconcave

u1(s1 ,s2)=½[4(s1 +s2 +bs1 s2)]- s12u2(s1 ,s2)=½[4(s1 +s2 +bs1 s2)]- s22

• Conclusion:thereexistsaNE!

• Ok,forthisgame,wealreadyknewit!• Butthetheoremismuchmoregeneralandappliestogameswherefindingtheequilibriumismuchmoredifficult

24

Onemorewordonthepartnershipgamebeforewemoveon

• Wehavefound(seeexercises)that– AtNashequilibrium:

s*1 =s*2 =1/(1-b)– Tomaximizethesumofutilities:

sW1 =sW2 =1/(1/2-b) >s*1• Sumofutilitiescalledsocialwelfare• BothpartnerswouldbebetteroffiftheyworkedsW1 (withsocialplanner,contract)

• Whydotheyworklessthanefficient?

25

Externality• Atthemargin,IbearthecostfortheextraunitofeffortIcontribute,butI’monlyreapinghalfoftheinducedprofits,becauseofprofitsharing

• Thisisknownasan“externality”èWhenI’mfiguringouttheeffortIhavetoputIdon’ttakeintoaccountthatotherhalfofprofitthatgoestomypartner

èInotherwords,myeffortbenefitsmypartner,notjustme

• Externalitiesareomnipresent:publicgoodproblems,freeriding,etc.(seemoreinthenetecon course)

26

Outline


27

Cournot Duopoly• Exampleofapplicationofgameswithcontinuousactionset

• Thisgameliesbetweentwoextremecasesineconomics,insituationswherefirms(e.g.twocompanies)arecompetingonthesamemarket– Perfectcompetition– Monopoly

• We’reinterestedinunderstandingwhathappensinthemiddle– Thegameanalysiswillgiveusinterestingeconomicinsightsontheduopolymarket

28

Cournot Duopoly:thegame• Theplayers:2Firms,e.g.,CokeandPepsi

• Strategies:quantitiesplayersproduceofidenticalproducts:qi,q-i– Productsareperfectsubstitutes

• Costofproduction:c*q– Simplemodelofconstantmarginalcost

• Prices:p=a– b(q1 +q2)=a– bQ– Market-clearingprice

29

PriceintheCournot duopoly

30

0

a

q1 +q2

p

Slope:-b

Demandcurve

Tellsthequantitydemandedforagivenprice

Cournot Duopoly:payoffs• Thepayoffs:firmsaimtomaximizeprofit

u1(q1,q2)=p*q1 – c*q1p=a– b(q1 +q2)

Øu1(q1,q2)=a*q1 – b*q21 – b*q1 q2 – c*q1

• Thegameissymmetric

Øu2(q1,q2)=a*q2 – b*q22 – b*q1 q2 – c*q231

Cournot Duopoly:bestresponses

02

02 21

<-

=---

b

cbqbqa• Firstordercondition

• Secondordercondition[make sure it’s a max]

è

ïïî

ïïí

ì

--

==

--

==

22)(ˆ

22)(ˆ

1122

2211

qbcaqBRq

qbcaqBRq

32

Cournot Duopoly:bestresponsediagramandNashequilibrium

33

0 q1

q2

bca

2-

bca -

NE

BR2

BR1

bcaqCournot 3

-=

bca -

bca

2-

Bestresponseatq2=0

• BR1(q2=0)=(a-c)/(2b)• Interpretation:monopolyquantity

Ømarginalrevenue=marginalcost

34

0 q1

p

DemandcurveSlope:-b

Marginalcost:c

MarginalrevenueSlope:-2b

bca

2-

a

MONOPOLY

WhenisBR1(q2)=0?

35

• BR1(q2=(a-c)/b)=0• Perfectcompetitionquantity

ØDemand=marginalcost

0 q1+q1

p

DemandcurveSlope:-b

Marginalcost

MarginalrevenueSlope:-2b

bca

2-

a

bca -

MONOPOLY PERFECTCOMPETITION

IfFirm1wouldproducemore,thesellingpricewouldnotcoverhercosts

Cournot Duopoly:bestresponsediagramandNashequilibrium

36

0 q1

q2

bca

2-

bca -

NE

BR2

BR1

bcaqCournot 3

-=Monopoly

Perfectcompetition

Strategicsubstitutes/complements

• InCournot duopoly:themoretheotherplayerdoes,thelessIwoulddo

è Thisisagameofstrategicsubstitutes– Note:ofcoursethegoodsweresubstitutes–We’retalkingaboutstrategieshere

• Inthepartnershipgame,itwastheopposite:themoretheotherplayerwouldthemoreIwoulddo

è Thisisagameofstrategiccomplements

37

Cournot duopoly:Marketperspective

• Totalindustryprofitmaximizedformonopoly

38

0 q1

q2

bca

2-

bca -

Industryprofitsaremaximized

BR2

BR1

bcaqCournot 3

-=

Cartel,agreement

• Howcouldthefirmssetanagreementtoincreaseprofit?

• Whatcantheproblemsbewiththisagreement?

390 q1

q2

bca

2-

bca -

BR2

BR1

bcaqCournot 3

-=Bothfirms

producehalfofthemonopolyquantity

Cournot Duopoly:lastobservations

• Howdoquantitiesandpriceswe’veencounteredsofarcompare?

PerfectCompetition

CournotQuantity Monopoly

MonopolyCournotQuantity

PerfectCompetition

bca -

bca

3)(2 -

bca

2-

QUANTITIES

PRICES

40

Summary

• Coordinationgames– ParetooptimalNEsometimesexist– Scopeforcommunication/leadership

• Gameswithcontinuousactionsets(purestrategies)– ComputeequilibriumwithFOC,SOC– Equilibriumexistsunderconcavityandcontinuityconditions

– Cournot duopoly

41

Game Theory -- Lecture 2 - EURECOM

Documents