GameTheory
-Lecture2
PatrickLoiseau
EURECOM
Fall2016
1
Lecture1recap
• Definedgamesinnormalform
• Defineddominancenotion
– Iterativedeletion
– Doesnotalwaysgiveasolution
• DefinedbestresponseandNashequilibrium
– ComputedNashequilibriuminsomeexamples
à AresomeNashequilibria betterthanothers?
à CanwealwaysfindaNashequilibrium?
2
Outline
1. CoordinationgamesandParetooptimality
2. Gameswithcontinuousactionsets
– Equilibriumcomputationandexistencetheorem
– Example:Cournot duopoly
3
Outline
1. CoordinationgamesandParetooptimality
2. Gameswithcontinuousactionsets
– Equilibriumcomputationandexistencetheorem
– Example:Cournot duopoly
4
TheInvestmentGame
• Theplayers:you
• Thestrategies:eachofyouchoosesbetweeninvesting
nothinginaclassproject($0)orinvesting($10)
• Payoffs:
– Ifyoudon’tinvestyourpayoffis$0
– Ifyouinvestyoumakeanetprofitof$5(grossprofit=
$15;investment$10) ifmorethan90%oftheclass
choosestoinvest.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
• Ifinvest$10à
$5netprofitif≥90%invest
-$10netprofitif<90%invest
• Let’splayagain!(nocommunication)
• Weareheadingtowardanequilibrium
èTherearecertaincasesinwhichplayingconvergesina
7
naturalsensetoanequilibrium
Paretodomination
• Isoneequilibriumbetterthantheother?
Definition: Paretodomination
AstrategyprofilesParetodominatesstrategy
profiles’iif foralli,ui(s)≥ui(s’)andthereexistsj
suchthatuj(s)>uj(s’);
i.e.,allplayershaveatleastashighpayoffsand
atleastoneplayerhasstrictlyhigherpayoff.
• Intheinvestmentgame?
8
Convergencetoequilibriuminthe
InvestmentGame
• WhydidweconvergetothewrongNE?
• Rememberwhenwestartedplaying
– Weweremoreorless50%investing
• Thestartingpointwasalreadybadforthepeople
whoinvestedforthemtoloseconfidence
• Thenwejusttumbleddown
• Whatwouldhavehappenedifwestartedwith
95%oftheclassinvesting?
9
Coordinationgame
• Thisisacoordinationgame
– We’dlikeeveryonetocoordinatetheiractionsandinvest
• Manyotherexamplesofcoordinationgames
–
–
–
–
PartyinaVilla
On-lineWebSites
Establishmentoftechnologicalmonopolies(Microsoft,HDTV)
Bankruns
• Unlikeinprisoner’sdilemma,communicationhelps in
coordinationgamesà scopeforleadership
– Inprisoner’sdilemma,atrustedthirdparty(TTP)wouldneedto
imposeplayerstoadoptastrictlydominatedstrategy
– Incoordinationgames,aTTPjustleadsthecrowdtowardsa
betterNEpoint(thereisnodominatedstrategy)
10
Battleofthesexes
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• FindtheNE
• IsthereaNEbetterthantheother(s)?
11
CoordinationGames
• Purecoordinationgames:thereisnoconflictwhether
oneNEisbetterthantheother
– E.g.:intheinvestmentgame,weallagreedthattheNE
witheveryoneinvestingwasa“better”NE
• Generalcoordinationgames:thereisasourceof
conflictasplayerswouldagreetocoordinate,butone
NEis“better”foraplayerandnotfortheother
– E.g.:BattleoftheSexes
è Communicationmightfailinthiscase
12
Paretooptimality
Definition: Paretooptimality
AstrategyprofilesisParetooptimalifthere
doesnotexistastrategyprofiles’thatPareto
dominatess.
• Battleofthesexes?
13
Outline
1. CoordinationgamesandParetooptimality
2. Gameswithcontinuousactionsets
– Equilibriumcomputationandexistencetheorem
– Example:Cournot duopoly
14
Thepartnershipgame(seeexercise
sheet2)
• 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(withtwice
continuouslydifferentiableutilities)
∂u1 (s1, s2 )
=0
∂s1
• Firstordercondition(FOC)
∂2 u1 (s1, s2 )
≤0
2
∂ s1
• Secondordercondition(SOC)
• Remark:theSOCisautomaticallysatisfiedifui(si,s-i)is
concaveinsi foralls-i (verystandardassumption)
• Remark2:becarefulwiththeborders!
– Exampleu1(s1,s2)=10-(s1+s2)2
– S1=[0,4],whatistheBRtos2=2?
– SolvingtheFOC,whatdoweget?
– WhentheFOCsolutionisoutsideSi,theBRisattheborder16
Nashequilibriumgraphically
s2
5
BR1(s2)
4
3
BR2(s1)
2
1
0
1
2
3
4
5
s1
• NEisfixedpointof(s1,s2)à (BR(s2),BR(s1))
17
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 ŝi ∈ BRi (s−i ) = argmax ui (si , s−i )
si
• Bestresponsecorrespondenceofi:s-i à BRi(s-i)
• (Correspondence=set-valuedfunction)
18
Nashequilibriumasafixedpoint
• Game ( N, ( Si )i∈N , (ui )i∈N )
• Let’sdefine(setofstrategyprofiles)
S = ×i∈N Si
andthecorrespondence
B:S →S
s  B(s) = ×i∈N BRi (s−i )
• Foragivens,B(s)isthesetofstrategyprofiless’
suchthatsi’isaBRtos-i foralli.
• Astrategyprofiles* isaNasheq.iif s* ∈ B(s* )
(justare-writingofthedefinition)
19
Kakutani’s fixedpointtheorem
Theorem: Kakutani’s fixedpointtheorem
LetX beacompactconvexsubsetofRn andlet
f : X → X beaset-valuedfunctionforwhich:
• forall,thesetisnonemptyconvex;
f (x)
x∈X
• thegraphoffisclosed.
Thenthereexistssuchthat
x * ∈ f (x * )
x* ∈ X
20
Closedgraph(upperhemicontinuity)
• Definition:fhasclosedgraphifforallsequences(xn)and(yn)
suchthatyn isinf(xn)foralln,xnàx andynày,yisinf(x)
• Alternativedefinition:fhasclosedgraphif forallxwehavethe
followingproperty:foranyopenneighborhoodVoff(x),there
existsaneighborhoodUofxsuchthatforallxinU,f(x)isa
subsetofV.
• Examples:
21
Existenceof(purestrategy)Nash
equilibrium
Theorem: ExistenceofpurestrategyNE
Supposethatthe gamesatisfies:
( N, (Si )i∈N , (ui )i∈N )
• Theactionsetofeachplayerisanonempty
Si
compactconvexsubsetofRn
ui
• Theutilityofeachplayeriscontinuousin
s
(on)andconcavein(on)
si
S
Si
Then,thereexistsa(purestrategy)Nash
equilibrium.
• Remark:theconcaveassumptioncanberelaxed
22
Proof
• DefineBasbefore.Bsatisfiestheassumptionsof
Kakutani’s fixedpointtheorem
• ThereforeBhasafixedpointwhichbydefinitionisa
Nashequilibrium!
• Now,weneedtoactuallyverifythatBsatisfiesthe
assumptionsofKakutani’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)]- s12
u2(s1 ,s2)=½[4(s1 +s2 +bs1 s2)]- s22
• Conclusion:thereexistsaNE!
• Ok,forthisgame,wealreadyknewit!
• Butthetheoremismuchmoregeneraland
appliestogameswherefindingtheequilibriumis
muchmoredifficult
24
Onemorewordonthepartnership
gamebeforewemoveon
• Wehavefound(seeexercises)that
– AtNashequilibrium:
s*1 =s*2 =1/(1-b)
– Tomaximizethesumofutilities:
sW1 =sW2 =1/(1/2-b) >s*1
• Sumofutilitiescalledsocialwelfare
• Bothpartnerswouldbebetteroffifthey
workedsW1 (withsocialplanner,contract)
• Whydotheyworklessthanefficient?
25
Externality
• Atthemargin,Ibearthecostfortheextraunitofeffort
Icontribute,butI’monlyreapinghalfoftheinduced
profits,becauseofprofitsharing
• Thisisknownasan“externality”
èWhenI’mfiguringouttheeffortIhavetoputIdon’t
takeintoaccountthatotherhalfofprofitthatgoesto
mypartner
èInotherwords,myeffortbenefitsmypartner,notjust
me
• Externalitiesareomnipresent:publicgoodproblems,
freeriding,etc.(seemoreinthenetecon course)
26
Outline
1. CoordinationgamesandParetooptimality
2. Gameswithcontinuousactionsets
– Equilibriumcomputationandexistencetheorem
– Example:Cournot duopoly
27
Cournot Duopoly
• Exampleofapplicationofgameswithcontinuous
actionset
• Thisgameliesbetweentwoextremecasesin
economics,insituationswherefirms(e.g.two
companies)arecompetingonthesamemarket
– Perfectcompetition
– Monopoly
• We’reinterestedinunderstandingwhathappens
inthemiddle
– Thegameanalysiswillgiveusinterestingeconomic
insightsontheduopolymarket
28
Cournot Duopoly:thegame
• Theplayers:2Firms,e.g.,CokeandPepsi
• Strategies:quantitiesplayersproduceofidentical
products:qi,q-i
– Productsareperfectsubstitutes
• Costofproduction:c*q
– Simplemodelofconstantmarginalcost
• Prices:p=a– b(q1 +q2)=a– bQ
– Market-clearingprice
29
PriceintheCournot duopoly
Demandcurve
p
Slope:-b
a
Tellsthequantity
demandedfora
givenprice
0
q1 +q2
30
Cournot Duopoly:payoffs
• Thepayoffs:firmsaimtomaximizeprofit
u1(q1,q2)=p*q1 – c*q1
p=a– b(q1 +q2)
Ø u1(q1,q2)=a*q1 – b*q21 – b*q1 q2 – c*q1
• Thegameissymmetric
Ø u2(q1,q2)=a*q2 – b*q22 – b*q1 q2 – c*q2
31
Cournot Duopoly:bestresponses
• Firstordercondition
a - 2bq1 - bq2 - c = 0
• Secondordercondition
- 2b < 0
[make sure it’s a max]
è
a - c q2
ì
ˆ
q
ïï 1 = BR1 (q2 ) = 2b - 2
í
ïqˆ = BR (q ) = a - c - q1
2
1
ïî 2
2b
2
32
Cournot Duopoly:bestresponse
diagramandNashequilibrium
q2
a-c
b
a-c
2b
BR1
NE
qCournot =
a-c
3b
BR2
0
a-c
2b
a-c
b
q1
33
Bestresponseatq2=0
p
a
• BR1(q2=0)=(a-c)/(2b)
• Interpretation:
Demandcurve
monopolyquantity
Slope:-b
Ømarginalrevenue=
Marginalrevenue
marginalcost
Slope:-2b
Marginalcost:c
0
a-c
2b
MONOPOLY
q1
34
p
a
WhenisBR1(q2)=0?
• BR1(q2=(a-c)/b)=0
• Perfectcompetition
Demandcurve
Slope:-b
quantity
ØDemand=marginal
Marginalrevenue
Slope:-2b
cost
Marginalcost
0
a-c
2b
MONOPOLY
a-c
b
PERFECT
COMPETITION
q1+q1
IfFirm1wouldproducemore,the
sellingpricewouldnotcoverhercosts
35
Cournot Duopoly:bestresponse
diagramandNashequilibrium
q2
a-c
b
BR1
Monopoly
NE
qCournot =
a-c
3b
BR2
0
a-c
2b
Perfect
competition
q1
36
Strategicsubstitutes/complements
• InCournot duopoly:themoretheotherplayer
does,thelessIwoulddo
è Thisisagameofstrategicsubstitutes
– Note:ofcoursethegoodsweresubstitutes
– We’retalkingaboutstrategieshere
• Inthepartnershipgame,itwastheopposite:
themoretheotherplayerwouldthemoreI
woulddo
è Thisisagameofstrategiccomplements
37
Cournot duopoly:Marketperspective
q2
• Totalindustry
profit
maximizedfor
monopoly
a-c
b
BR1
Industryprofits
aremaximized
0
qCournot =
a-c
2b
a-c
3b
BR2
q1
38
Cartel,agreement
q2
a-c
b
BR1
Bothfirms
producehalf
ofthemonopoly
quantity
qCournot =
0
a-c
2b
a-c
3b
BR2
• Howcouldthe
firmssetan
agreementto
increaseprofit?
• Whatcanthe
problemsbe
withthis
agreement?
q1
39
Cournot Duopoly:lastobservations
• Howdoquantitiesandpriceswe’ve
encounteredsofarcompare?
Perfect
Competition
Cournot
Quantity
Monopoly
a-c
b
2(a - c)
3b
a-c
2b
Perfect
Competition
Cournot
Quantity
Monopoly
QUANTITIES
PRICES
40
Summary
• Coordinationgames
– ParetooptimalNEsometimesexist
– Scopeforcommunication/leadership
• Gameswithcontinuousactionsets(pure
strategies)
– ComputeequilibriumwithFOC,SOC
– Equilibriumexistsunderconcavityandcontinuity
conditions
– Cournot duopoly
41