game theory -- lecture 2 - eurecom
TRANSCRIPT
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
1. CoordinationgamesandParetooptimality2. Gameswithcontinuousactionsets– Equilibriumcomputationandexistencetheorem– Example:Cournot duopoly
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
1. CoordinationgamesandParetooptimality2. Gameswithcontinuousactionsets– Equilibriumcomputationandexistencetheorem– Example:Cournot duopoly
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
1. CoordinationgamesandParetooptimality2. Gameswithcontinuousactionsets– Equilibriumcomputationandexistencetheorem– Example:Cournot duopoly
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