GameTheory
-Lecture1
PatrickLoiseau
EURECOM
Fall2016
1
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
2
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
3
Let’splaythe“gradegame”
Withoutshowingyourneighborswhatyouaredoing,writedownon
aformeithertheletteralpha ortheletterbeta.Thinkofthisasa
“gradebid”.Iwillrandomlypairyourformwithoneotherform.
Neitheryounoryourpairwilleverknowwithwhomyouwere
paired.Hereishowgradesmaybeassignedforthisclass:
• Ifyouputalpha andyourpairputsbeta,thenyouwillgetgradeA,
andyourpairgradeC;
• Ifbothyouandyourpairputalpha,thenyoubothwillgetthe
gradeB-;
• Ifyouputbeta andyourpairputsalpha,thenyouwillgetthegrade
CandyourpairgradeA;
• Ifbothyouandyourpairputbeta,thenyouwillbothgetgradeB+
4
Whatisgametheory?
• Gametheoryisamethodofstudyingstrategic
situations,i.e.,wheretheoutcomesthataffectyou
dependonactionsofothers,notonlyyours
• Informally:
– AtoneendwehaveFirmsinperfectcompetition:inthis
case,firmsarepricetakersanddonotcareaboutwhat
otherdo
– AttheotherendwehaveMonopolistFirms:inthiscase,a
firmdoesn’thavecompetitorstoworryabout,they’renot
price-takersbuttheytakethedemandcurve
– Everythinginbetweenisstrategic,i.e.,everythingthat
constitutesimperfectcompetition
• Example:Theautomotiveindustry
• Gametheoryhasbecomeamultidisciplinaryarea
– Economics,mathematics,computerscience,engineering…5
Outcomematrix
• Justreadingthetextishardtoabsorb,let’s
useaconcisewayofrepresentingthegame:
mypair
alpha
alpha
mypair
beta
B-
alpha
A
me
beta
alpha
B-
C
beta
A
B+
me
beta
C
B+
mygrades
pair’sgrades
6
Outcomematrix(2)
• Weuseamorecompactrepresentation:
mypair
alpha
beta
Thisisanoutcomematrix:
alpha
B- ,B-
beta
C,A
A,C
Ittellsuseverythingthatwas
inthegamewesaw
me
1st grade:rowplayer
(mygrade)
B+,B+
2nd grade:columnplayer
(mypair’sgrade)
7
Thegradegame:discussion
• Whatdidyouchoose?Why?
• Twopossiblewayofthinking:
– Regardlessofmypartnerchoice,therewouldbebetter
outcomesformebychoosingalpharatherthanbeta;
– Wecouldallbecollusiveandworktogether,henceby
choosingbetawewouldgethighergrades.
• Wedon’thaveagameyet!
– Wehaveplayers andstrategies (i.e.,possibleactions)
– Wearemissingobjectives
• Objectivescanbedefinedintwoways
– Preferences,i.e.,orderingofpossibleoutcomes
– Payoffs orutility functions
8
Thegradegame:payoffmatrix
• Possiblepayoffs:inthiscaseweonlycare
aboutourowngrades
mypair
alpha
beta
alpha
0,0
3,-1
beta
-1,3
1,1
#ofutiles,orutility:
(A,C)à 3
(B-,B-)à 0
Hencethepreferenceorderis:
me
A>B+>B- >C
• Howtochooseanactionhere?
9
Strictlydominatedstrategies
• Playalpha!
– Indeed,nomatterwhatthepairdoes,byplaying
alphayouwouldobtainahigherpayoff
Definition:
Wesaythatmystrategyalphastrictlydominates
mystrategybeta,ifmypayofffromalphais
strictlygreater thanthatfrombeta,regardlessof
whatothersdo.
à Donotplayastrictlydominated strategy!
10
Rationalchoiceoutcome
• Ifwe(meandmypair)reasonselfishly,wewillbothselectalpha,
andgetapayoffof0;
• Butwecouldendupboth withapayoffof1…
• What’stheproblemwiththis?
– Supposeyouhavesupermentalpowerandobligeyourpartnerto
agreewithyouandchoosebeta,sothatyoubothwouldendupwitha
payoffof1…
– Evenwithcommunication,itwouldn’twork,becauseatthispoint,
you’dbebetterofbychoosingalpha,andgetapayoffof3
à Rationalchoice(i.e.,notchoosingadominatedstrategy)canlead
tobadoutcomes!
• Solutions?
– Contracts,treaties,regulations: changepayoff
– Repeatedplay
11
Theprisoner’sdilemma
Prisoner2
• Importantclassofgames
• Otherexamples
D
C
1. Jointproject:
•
Eachindividualmayhavean
incentivetoshirk
2. Pricecompetition
•
•
D
-5,-5
0,-6
Prisoner1
Eachfirmhasanincentiveto
undercutprices
Ifallfirmsbehavethisway,
pricesaredrivendowntowards
marginalcostandindustryprofit
willsuffer
C
-6,0
-2,-2
3. Commonresource
•
•
Carbonemissions
Fishing
12
Anotherpossiblepayoffmatrix
• Thistimepeoplearemoreinclinetobealtruistic
mypair
#ofutiles,orutility:
(A,C)à 3– 4=-1
my‘A’- myguilt
(C,A)à -1– 2=-3
my‘C’- myindignation
Thisisacoordinationproblem
• Whatwouldyouchoosenow?
alpha
beta
alpha
0,0
-1,-3
beta
-3,-1
1,1
me
– Nodominatedstrategy
à Payoffsmatter.(wewillcomebacktothisgamelater)
13
Anotherpossiblepayoffmatrix(2)
• Selfishvs.Altruistic
• Whatdoyouchoose?
mypair
(Altruistic)
alpha
beta
0,0
3,-3
-1,-1
1,1
Inthiscase,alphastilldominates
ThefactI(selfishplayer)amplaying
againstanaltruisticplayerdoesn’tchange
mystrategy,evenbychangingtheother
Player’spayoff
alpha
Me
(Selfish)
beta
14
Anotherpossiblepayoffmatrix(3)
• Altruisticvs.Selfish
• Whatdoyouchoose?
•DoIhaveadominatingstrategy?
•Doestheotherplayerhaveadominating
strategy?
Bythinkingofwhatmy“opponent”willdo
Icandecidewhattodo.
mypair
(Selfish)
alpha
alpha
beta
0,0
-1,-1
-3,3
1,1
Me
(Altruistic)
beta
à Putyourselfinotherplayers’shoesandtry
tofigureoutwhattheywilldo
15
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
16
Gameinnormalform
Notation
E.g.:gradegame
Players
i,j,…
Me andmypair
Strategies
si:aparticularstrategyof
playeri
alpha
s-i:the strategyof
everybodyelseexcept
playeri
Payoffs
Si:the setofpossible
strategiesofplayeri
{alpha,beta}
s:aparticularplayofthe
game
“strategy profile”
(vector,orlist)
(alpha,alpha)
ui(s1,…,si,…,sN)=ui(s)
ui(s)= seepayoffmatrix
17
Assumptions
• Weassumealltheingredientsofthegameto
beknown
– Everybodyknowsthepossiblestrategieseveryone
elsecouldchoose
– Everybodyknowseveryoneelse’spayoffs
• Thisisnotveryrealistic,butthingsare
complicatedenoughtogiveusmaterialfor
thisclass
18
Strictdominance
Definition: Strict dominance
Wesayplayeri’s strategy si’isstrictly dominated
byplayeri’s strategysi if:
ui(si,s-i)>ui(si’,s-i)forall s-i
Nomatterwhatotherpeopledo, bychoosingsi
insteadofsi’,playeri willalwaysobtainahigher
payoff.
19
Example1
2
C
L
1
T
B
5,-1
6,4
11,3
0,2
R
0,0
2,0
Players
1,2
Strategysets
S1={T,B}
S2={L,C,R}
Payoffs
U1(T,C)=11
U2(T,C) =3
NOTE:Thisgameisnotsymmetric
20
Example2:“Hannibal”game
• Aninvaderisthinkingaboutinvadingacountry,and
thereare2waysthroughwhichhecanleadhisarmy.
• Youarethedefenderofthiscountryandyouhaveto
decidewhichofthesewaysyouchoosetodefend:you
canonlydefendoneoftheseroutes.
• Onerouteisahardpass:iftheinvaderchoosesthis
routehewillloseonebattalionofhisarmy(overthe
mountains).
• Iftheinvadermeetsyourarmy,whateverroutehe
chooses,hewillloseabattalion
21
Example2:“Hannibal”game
attacker
e
defender
E
H
1,1
0,2
h
1,1
2,0
e,E=easy;h,H =hard
• Attacker’spayoffsishowmanybattalionshe
willarrivewithinyourcountry
– Defender’spayoffisthecomplementaryto2
• Youarethedefender,whatdoyoudo?
22
Weakdominance
Definition: Weakdominance
Wesayplayeri’sstrategy si’isweakly
dominatedbyplayeri’sstrategysi if:
ui(si,s-i)≥ui(si’,s-i)forall s-i
ui(si,s-i)>ui(si’,s-i)forsome s-i
Nomatterwhatotherpeopledo, bychoosingsi
insteadofsi’,playeri willalwaysobtainapayoff
atleastashighandsometimeshigher.
23
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
24
The“PickaNumber”Game
Withoutshowingyourneighborwhatyou’redoing,write
downanintegernumberbetween1and100.Iwillcalculate
theaveragenumberchosenintheclass.Thewinnerinthis
gameisthepersonwhosenumberisclosesttotwo-thirdsof
theaverageintheclass.Thewinnerwillwin5eurominus
thedifferenceincentsbetweenherchoiceandthattwothirdsoftheaverage.
Example:3students
Numbers:25,5,60
Total:90,Average:30,2/3*average:20
25wins:5euro– 5cents=4.95euro
25
Firstreasoning
• Apossibleassumption:
– Peoplechosenumbersuniformlyatrandom
èTheaverageis50
è2/3*average=33.3
• What’swrongwiththisreasoning?
26
Rationality:dominatedstrategies
• Aretheredominatedstrategies?
• Ifeveryonewouldchose100,thenthe
winningnumberwouldbe66
ènumbers>67areweaklydominatedby66
èRationalitytellsnottochoosenumbers>67
27
Knowledgeofrationality
• Sonowwe’veeliminateddominatedstrategies,
it’slikethegamewastobeplayedovertheset[1,
…,67]
• Onceyoufiguredoutthatnobodyisgoingto
choseanumberabove67,theconclusionis
èAlsostrategiesabove45areruledout
èTheyareweaklydominated,onlyoncewedelete
68-100
• Thisimpliesrationality,andknowledgethat
othersarerationalaswell
28
Commonknowledge
• Commonknowledge:youknowthatothersknow
thatothersknow…andsoonthatrationalityis
underlyingallplayers’choices
• …1wasthewinningstrategy!!
• Inpractice:
– Averagewas:Winningwas:2/3*average
• Nowlet’splayagain!
29
Warningoniterativedeletion
• Iterativedeletionofdominatedstrategies
seemsapowerfulidea,butit’salsodangerous
ifyoutakeitliterally
• Insomegames,iterativedeletionconvergesto
asinglechoice,inothersitmaynot(see
Osborne-Rubinstein)
30
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
31
Asimplemodelinpolitics
• 2candidates choosingtheirpoliticalpositions
onaspectrum
• Assumethespectrumhas10positions,with
10%votersoneach
• Assumevotersvoteforclosestcandidateand
breaktiesbysplittingvotesequally
• Candidate’spayoff=shareofvotes
1
LEFTWING
2
3
4
5
6
7
8
9
10
RIGHTWING
32
Dominatedstrategies
• Isposition1dominated?
– Testingdominationby2
Vs.1
u1(1,1)=50%
<
u1(2,1)=90%
Vs.2
u1(1,2)=10%
<
u1(2,2)=50%
Vs.3
u1(1,3)=15%
<
u1(2,3)=20%
Vs.4
u1(1,4)=20%
<
u1(2,4)=25%
…
…
…
….
• Samereasoningà 9strictlydominates10
33
Otherdominatedstrategies?
• Is2dominatedby3?
• Canwegofurther?
34
TheMedianVoterTheorem
• Continuingtheprocessofiterativedeletion
– Onlypositions5and6remain
èCandidateswillbesqueezedtowardsthecenter,
i.e.,theywillchoosepositionsveryclosetoeach
other
Inpoliticalsciencethisiscalledthe
MedianVoterTheorem
35
TheMedianVoterTheorem
• Otherapplicationineconomics:product
placement
• Example:
– Youareplacingagasstation
– youmightthinkthatitwouldbeniceifgasstations
spreadthemselvesevenlyoutoverthetown,oron
everyroad,sothattherewouldbeastationcloseby
whenyourunoutofgas
• Asweallknow,thisdoesn’thappen:allgas
stationstendtocrowdintothesamecorners,all
thefastfoodscrowdaswell, etc.
36
Critics
• Weusedamodelofareal-worldsituation,andtriedto
predicttheoutcomeusinggametheory
• Themodelissimplified:itmissesmanyfeatures!
– Votersarenotevenlydistributed
– Manyvotersdonotvote
– Theremaybemorethan2candidates
• Soisthismodel(andmodelingingeneral)useless?
• No!First,analyzeaproblemwithsimplifyingassumptions,
thenrelaxthemandseewhathappens
– E.g.:wouldadifferentvotersdistributionchangetheresult?
• Wewillseethroughoutthecourse(andintheNetEcon
course)examplesofsimplifiedmodelgivingveryuseful
predictions
37
Lecture1outline
1. Introduction
2. Definitionsandnotation
– Gameinnormalform
– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy
– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
38
Example
U
Player1 M
D
l
Player2
c
0,4
4,0
3,5
4,0
0,4
3,5
r
5,3
5,3
6,6
•
Isthereanydominated strategyforplayer1/2?
•
Whatwouldplayer1doifplayer2plays
– left?
– center?
– right?
•
Whatwouldplayer2doifplayer1plays
– Up?
– Middle?
– Down?
39
Bestresponsedefinition
Definition: Best Response
Playeri’s strategyŝi isaBRtostrategys-i ofother
playersif:
ui(ŝi ,s-i)≥ui(s’i ,s-i)for alls’i inSi
or
ŝi solves max ui(si ,s-i)
40
Bestresponsesinthesimplegame
U
Player1 M
D
l
Player2
c
0,4
4,0
3,5
4,0
0,4
3,5
r
5,3
5,3
6,6
• BR1(l)=M BR2(U)=l
• BR1(c)=U BR2(M)=c
• BR1(r)=D BR2(D)=r
• Doesthissuggestasolutionconcept?
41
Nashequilibriumdefinition
Definition: NashEquilibrium
Astrategyprofile(s1*,s2*,…,sN*)isaNash
Equilibrium(NE)if,foreachi,herchoicesi*isa
bestresponsetotheotherplayers’choicess-i*
• Onofthemostimportantconceptingame
theory
– Usedinmanyapplications
• SeminalpaperJ.Nash(1951)
– Nobel1994
42
Nashequilibriuminthesimplegame
U
Player1 M
D
l
Player2
c
0,4
4,0
3,5
4,0
0,4
3,5
r
5,3
5,3
6,6
• BR1(l)=M BR2(U)=l
• BR1(c)=U BR2(M)=c
• BR1(r)=D BR2(D)=r
• (D,r)isaNE
43
NEmotivation
• Realplayersdon’talwaysplayNEbut
• Noregret:Holdingeveryoneelse’sstrategiesfixed,no
individualhasastrict incentivetomoveaway
– Havingplayedagame,supposeyouplayedaNE:lookingback
theanswertothequestion“DoIregretmyactions?”wouldbe
“No,givenwhatotherplayersdid,Ididmybest”
– Sometimesusedasadefinition:aNEisaprofilesuchthatno
playercanstrictlyimprovebyunilateraldeviation
• Self-fulfillingbelief:
– IfIbelieveeveryoneisgoingtoplaytheirpartsofaNE,then
everyonewillinfactplayaNE
• Wewillseeothermotivations
44
Remark:Bestresponsemaynotbe
unique
U
Player1 M
D
l
Player2
c
0,2
11,1
0,3
2,3
3,2
1,0
r
4,3
0,0
8,0
• Findallbestresponses
• FindNE
45
NEvs.strictdominance
Player2
Player1
alpha
beta
alpha
beta
0,0
-1,3
3,-1
1,1
• Whatisthisgame?
• FindNEanddominatedstrategies.
èNostrictlydominatedstrategiescouldeverbe
playedinNE
– Indeed,astrictlydominatedstrategyisneverabest
responsetoanything
46
NEvs.weakdominance
• Canaweaklydominatedstrategybeplayedin
NE?
Player2
r
l
• Example:
Player1
U
D
1,1
0,0
0,0
0,0
• Arethereanydominatedstrategies?
• FindNE
• Conclude
47
Summaryoflecture1
• Basicconceptsseeninthislecture
– Gameinnormalform
– Dominatedstrategies(strict,weak),iterativedeletion
– Bestresponse andNashequilibrium
• Gametheoryisamathematicaltooltostudy
strategicinteractions,i.e.,situationswherean
agent’soutcomedependsnotonlyonhisown
actionbutalsoonotheragents’actions
– Manyapplications(wewillseesome)
– Understandtheworld
48
Remark
• Inmostofthegamesseeninthislecture,the
actionsetswerefinite(i.e.,playershadafinite
numberofactionstochoosefrom)
• Thisisnotageneralthing:wewillseemany
gameswithcontinuousactionsets(exercises
andnextlectures)
– Example:companieschoosingprices
49