GameTheory--
Lecture1
PatrickLoiseauEURECOMFall2016
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Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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Let’splaythe“gradegame”Withoutshowingyourneighborswhatyouaredoing,writedownonaformeithertheletteralpha ortheletterbeta.Thinkofthisasa“gradebid”.Iwillrandomlypairyourformwithoneotherform.Neitheryounoryourpairwilleverknowwithwhomyouwerepaired.Hereishowgradesmaybeassignedforthisclass:
• Ifyouputalpha andyourpairputsbeta,thenyouwillgetgradeA,andyourpairgradeC;
• Ifbothyouandyourpairputalpha,thenyoubothwillgetthegradeB-;
• Ifyouputbeta andyourpairputsalpha,thenyouwillgetthegradeCandyourpairgradeA;
• Ifbothyouandyourpairputbeta,thenyouwillbothgetgradeB+
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Whatisgametheory?• Gametheoryisamethodofstudyingstrategicsituations,i.e.,wheretheoutcomesthataffectyoudependonactionsofothers,notonlyyours
• Informally:– AtoneendwehaveFirmsinperfectcompetition:inthiscase,firmsarepricetakersanddonotcareaboutwhatotherdo
– AttheotherendwehaveMonopolistFirms:inthiscase,afirmdoesn’thavecompetitorstoworryabout,they’renotprice-takersbuttheytakethedemandcurve
– Everythinginbetweenisstrategic,i.e.,everythingthatconstitutesimperfectcompetition• Example:Theautomotiveindustry
• Gametheoryhasbecomeamultidisciplinaryarea– Economics,mathematics,computerscience,engineering…5
Outcomematrix
• Justreadingthetextishardtoabsorb,let’suseaconcisewayofrepresentingthegame:
alpha beta
alpha
beta
B- A
B+C
me
mypairalpha beta
alpha
beta
B- C
B+A
me
mypair
mygrades pair’sgrades 6
Outcomematrix(2)
• Weuseamorecompactrepresentation:
alpha beta
alpha
beta
B- ,B- A,C
B+,B+C,A
me
mypair
1st grade:rowplayer(mygrade)
2nd grade:columnplayer(mypair’sgrade)
Thisisanoutcomematrix:
Ittellsuseverythingthatwasinthegamewesaw
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Thegradegame:discussion• Whatdidyouchoose?Why?• Twopossiblewayofthinking:– Regardlessofmypartnerchoice,therewouldbebetteroutcomesformebychoosingalpharatherthanbeta;
– Wecouldallbecollusiveandworktogether,hencebychoosingbetawewouldgethighergrades.
• Wedon’thaveagameyet!– Wehaveplayers andstrategies (i.e.,possibleactions)– Wearemissingobjectives
• Objectivescanbedefinedintwoways– Preferences,i.e.,orderingofpossibleoutcomes– Payoffs orutility functions
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Thegradegame:payoffmatrix
• Possiblepayoffs:inthiscaseweonlycareaboutourowngrades
• Howtochooseanactionhere?
alpha beta
alpha
beta
0,0 3,-1
1,1-1,3
me
mypair
#ofutiles,orutility:
(A,C)à 3
(B-,B-)à 0
Hencethepreferenceorderis:
A>B+>B- >C
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Strictlydominatedstrategies• Playalpha!– Indeed,nomatterwhatthepairdoes,byplayingalphayouwouldobtainahigherpayoff
Definition:Wesaythatmystrategyalphastrictlydominatesmystrategybeta,ifmypayofffromalphaisstrictlygreater thanthatfrombeta,regardlessofwhatothersdo.
à Donotplayastrictlydominated strategy!
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Rationalchoiceoutcome• Ifwe(meandmypair)reasonselfishly,wewillbothselectalpha,
andgetapayoffof0;• Butwecouldendupboth withapayoffof1…• What’stheproblemwiththis?
– Supposeyouhavesupermentalpowerandobligeyourpartnertoagreewithyouandchoosebeta,sothatyoubothwouldendupwithapayoffof1…
– Evenwithcommunication,itwouldn’twork,becauseatthispoint,you’dbebetterofbychoosingalpha,andgetapayoffof3
à Rationalchoice(i.e.,notchoosingadominatedstrategy)canleadtobadoutcomes!
• Solutions?– Contracts,treaties,regulations: changepayoff– Repeatedplay
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Theprisoner’sdilemma• Importantclassofgames• Otherexamples
1. Jointproject:• Eachindividualmayhavean
incentivetoshirk2. Pricecompetition
• Eachfirmhasanincentivetoundercutprices
• Ifallfirmsbehavethisway,pricesaredrivendowntowardsmarginalcostandindustryprofitwillsuffer
3. Commonresource• Carbonemissions• Fishing
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D C
D
C
-5,-5 0,-6
-2,-2-6,0
Prisoner1
Prisoner2
Anotherpossiblepayoffmatrix• Thistimepeoplearemoreinclinetobealtruistic
• Whatwouldyouchoosenow?– Nodominatedstrategy
à Payoffsmatter.(wewillcomebacktothisgamelater)
alpha beta
alpha
beta
0,0 -1,-3
1,1-3,-1
me
mypair#ofutiles,orutility:
(A,C)à 3– 4=-1my‘A’- myguilt
(C,A)à -1– 2=-3my‘C’- myindignation
Thisisacoordinationproblem
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Anotherpossiblepayoffmatrix(2)
• Selfishvs.Altruistic• Whatdoyouchoose?
alpha beta
alpha
beta
0,0 3,-3
1,1-1,-1
Me(Selfish)
mypair(Altruistic)
Inthiscase,alphastilldominates
ThefactI(selfishplayer)amplayingagainstanaltruisticplayerdoesn’tchangemystrategy,evenbychangingtheotherPlayer’spayoff
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Anotherpossiblepayoffmatrix(3)
• Altruisticvs.Selfish• Whatdoyouchoose?
à Putyourselfinotherplayers’shoesandtrytofigureoutwhattheywilldo
alpha beta
alpha
beta
0,0 -1,-1
1,1-3,3
Me(Altruistic)
mypair(Selfish)
•DoIhaveadominatingstrategy?•Doestheotherplayerhaveadominatingstrategy?
Bythinkingofwhatmy“opponent”willdoIcandecidewhattodo.
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Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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Gameinnormalform
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Notation E.g.:gradegame
Players i,j,… Me andmypair
Strategies si:aparticularstrategyofplayeri
s-i:the strategyofeverybodyelseexceptplayeri
alpha
Si:the setofpossiblestrategiesofplayeri
{alpha,beta}
s:aparticularplayofthegame“strategy profile”(vector,orlist)
(alpha,alpha)
Payoffs ui(s1,…,si,…,sN)=ui(s) ui(s)= seepayoffmatrix
Assumptions
• Weassumealltheingredientsofthegametobeknown– Everybodyknowsthepossiblestrategieseveryoneelsecouldchoose
– Everybodyknowseveryoneelse’spayoffs
• Thisisnotveryrealistic,butthingsarecomplicatedenoughtogiveusmaterialforthisclass
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Strictdominance
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Definition: Strict dominanceWesayplayeri’s strategy si’isstrictly dominatedbyplayeri’s strategysi if:
ui(si,s-i)>ui(si’,s-i)forall s-i
Nomatterwhatotherpeopledo, bychoosingsiinsteadofsi’,playeri willalwaysobtainahigherpayoff.
Example1
5,-1 11,3 0,06,4 0,2 2,0
T
B
L C R
1
2
Players 1,2
Strategysets S1={T,B} S2={L,C,R}
Payoffs U1(T,C)=11 U2(T,C) =3
NOTE:Thisgameisnotsymmetric20
Example2:“Hannibal”game• Aninvaderisthinkingaboutinvadingacountry,andthereare2waysthroughwhichhecanleadhisarmy.
• Youarethedefenderofthiscountryandyouhavetodecidewhichofthesewaysyouchoosetodefend:youcanonlydefendoneoftheseroutes.
• Onerouteisahardpass:iftheinvaderchoosesthisroutehewillloseonebattalionofhisarmy(overthemountains).
• Iftheinvadermeetsyourarmy,whateverroutehechooses,hewillloseabattalion
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Example2:“Hannibal”game
e,E=easy;h,H =hard
• Attacker’spayoffsishowmanybattalionshewillarrivewithinyourcountry– Defender’spayoffisthecomplementaryto2
• Youarethedefender,whatdoyoudo?
1,1 1,10,2 2,0
E
H
e h
defender
attacker
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Weakdominance
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Definition:WeakdominanceWesayplayeri’sstrategy si’isweaklydominatedbyplayeri’sstrategysi if:
ui(si,s-i)≥ui(si’,s-i)forall s-iui(si,s-i)>ui(si’,s-i)forsome s-i
Nomatterwhatotherpeopledo, bychoosingsiinsteadofsi’,playeri willalwaysobtainapayoffatleastashighandsometimeshigher.
Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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The“PickaNumber”GameWithoutshowingyourneighborwhatyou’redoing,writedownanintegernumberbetween1and100.Iwillcalculatetheaveragenumberchosenintheclass.Thewinnerinthisgameisthepersonwhosenumberisclosesttotwo-thirdsoftheaverageintheclass.Thewinnerwillwin5eurominusthedifferenceincentsbetweenherchoiceandthattwo-thirdsoftheaverage.
Example:3studentsNumbers:25,5,60Total:90,Average:30,2/3*average:20
25wins:5euro– 5cents=4.95euro
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Firstreasoning
• Apossibleassumption:– PeoplechosenumbersuniformlyatrandomèTheaverageis50è2/3*average=33.3
• What’swrongwiththisreasoning?
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Rationality:dominatedstrategies
• Aretheredominatedstrategies?• Ifeveryonewouldchose100,thenthewinningnumberwouldbe66
ènumbers>67areweaklydominatedby66èRationalitytellsnottochoosenumbers>67
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Knowledgeofrationality
• Sonowwe’veeliminateddominatedstrategies,it’slikethegamewastobeplayedovertheset[1,…,67]
• Onceyoufiguredoutthatnobodyisgoingtochoseanumberabove67,theconclusionis
èAlsostrategiesabove45areruledoutèTheyareweaklydominated,onlyoncewedelete68-100
• Thisimpliesrationality,andknowledgethatothersarerationalaswell 28
Commonknowledge
• Commonknowledge:youknowthatothersknowthatothersknow…andsoonthatrationalityisunderlyingallplayers’choices
• …1wasthewinningstrategy!!
• Inpractice:– Averagewas:Winningwas:2/3*average
• Nowlet’splayagain!29
Warningoniterativedeletion
• Iterativedeletionofdominatedstrategiesseemsapowerfulidea,butit’salsodangerousifyoutakeitliterally
• Insomegames,iterativedeletionconvergestoasinglechoice,inothersitmaynot(seeOsborne-Rubinstein)
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Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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Asimplemodelinpolitics
• 2candidates choosingtheirpoliticalpositionsonaspectrum
• Assumethespectrumhas10positions,with10%votersoneach
• Assumevotersvoteforclosestcandidateandbreaktiesbysplittingvotesequally
• Candidate’spayoff=shareofvotes
1 2 3 4 5 6 7 8 9 10
LEFTWING RIGHTWING 32
Dominatedstrategies
• Isposition1dominated?– Testingdominationby2
• Samereasoningà 9strictlydominates10
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%… … … ….
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Otherdominatedstrategies?
• Is2dominatedby3?
• Canwegofurther?
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TheMedianVoterTheorem
• Continuingtheprocessofiterativedeletion– Onlypositions5and6remain
èCandidateswillbesqueezedtowardsthecenter,i.e.,theywillchoosepositionsveryclosetoeachother
InpoliticalsciencethisiscalledtheMedianVoterTheorem
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TheMedianVoterTheorem• Otherapplicationineconomics:productplacement
• Example:– Youareplacingagasstation– youmightthinkthatitwouldbeniceifgasstationsspreadthemselvesevenlyoutoverthetown,oroneveryroad,sothattherewouldbeastationclosebywhenyourunoutofgas
• Asweallknow,thisdoesn’thappen:allgasstationstendtocrowdintothesamecorners,allthefastfoodscrowdaswell, etc.
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Critics• Weusedamodelofareal-worldsituation,andtriedto
predicttheoutcomeusinggametheory• Themodelissimplified:itmissesmanyfeatures!
– Votersarenotevenlydistributed– Manyvotersdonotvote– Theremaybemorethan2candidates
• Soisthismodel(andmodelingingeneral)useless?• No!First,analyzeaproblemwithsimplifyingassumptions,
thenrelaxthemandseewhathappens– E.g.:wouldadifferentvotersdistributionchangetheresult?
• Wewillseethroughoutthecourse(andintheNetEconcourse)examplesofsimplifiedmodelgivingveryusefulpredictions 37
Lecture1outline
1. Introduction2. Definitionsandnotation– Gameinnormalform– Strictandweakdominance
3. Iterativedeletionofdominatedstrategy– Afirstmodelinpolitics
4. BestresponseandNashequilibrium
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Example
• Isthereanydominated strategyforplayer1/2?
• Whatwouldplayer1doifplayer2plays– left?– center?– right?
• Whatwouldplayer2doifplayer1plays– Up?– Middle?– Down?
0,4 4,0 5,34,0 0,4 5,33,5 3,5 6,6
U
M
l r
Player1
Player2
D
c
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Bestresponsedefinition
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Definition: Best ResponsePlayeri’s strategyŝi isaBRtostrategys-i ofotherplayersif:
ui(ŝi ,s-i)≥ui(s’i ,s-i)for alls’i inSior
ŝi solvesmax ui(si ,s-i)
Bestresponsesinthesimplegame
• BR1(l)=M BR2(U)=l• BR1(c)=U BR2(M)=c• BR1(r)=D BR2(D)=r
• Doesthissuggestasolutionconcept?
0,4 4,0 5,34,0 0,4 5,33,5 3,5 6,6
U
M
l r
Player1
Player2
D
c
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Nashequilibriumdefinition
• Onofthemostimportantconceptingametheory– Usedinmanyapplications
• SeminalpaperJ.Nash(1951)– Nobel1994
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Definition: NashEquilibriumAstrategyprofile(s1*,s2*,…,sN*)isaNashEquilibrium(NE)if,foreachi,herchoicesi*isabestresponsetotheotherplayers’choicess-i*
Nashequilibriuminthesimplegame
• BR1(l)=M BR2(U)=l• BR1(c)=U BR2(M)=c• BR1(r)=D BR2(D)=r
• (D,r)isaNE
0,4 4,0 5,34,0 0,4 5,33,5 3,5 6,6
U
M
l r
Player1
Player2
D
c
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NEmotivation• Realplayersdon’talwaysplayNEbut
• Noregret:Holdingeveryoneelse’sstrategiesfixed,noindividualhasastrict incentivetomoveaway– Havingplayedagame,supposeyouplayedaNE:lookingback
theanswertothequestion“DoIregretmyactions?”wouldbe“No,givenwhatotherplayersdid,Ididmybest”
– Sometimesusedasadefinition:aNEisaprofilesuchthatnoplayercanstrictlyimprovebyunilateraldeviation
• Self-fulfillingbelief:– IfIbelieveeveryoneisgoingtoplaytheirpartsofaNE,then
everyonewillinfactplayaNE
• Wewillseeothermotivations44
Remark:Bestresponsemaynotbeunique
• Findallbestresponses
• FindNE
0,2 2,3 4,311,1 3,2 0,00,3 1,0 8,0
U
M
l r
Player1
Player2
D
c
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NEvs.strictdominance
• Whatisthisgame?• FindNEanddominatedstrategies.
èNostrictlydominatedstrategiescouldeverbeplayedinNE– Indeed,astrictlydominatedstrategyisneverabestresponsetoanything
0,0 3,-1-1,3 1,1
alpha
beta
alpha
Player1
Player2beta
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NEvs.weakdominance
• CanaweaklydominatedstrategybeplayedinNE?
• Example:
• Arethereanydominatedstrategies?• FindNE• Conclude
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1,1 0,00,0 0,0
U
D
l
Player1
Player2r
Summaryoflecture1
• Basicconceptsseeninthislecture– Gameinnormalform– Dominatedstrategies(strict,weak),iterativedeletion– Bestresponse andNashequilibrium
• Gametheoryisamathematicaltooltostudystrategicinteractions,i.e.,situationswhereanagent’soutcomedependsnotonlyonhisownactionbutalsoonotheragents’actions– Manyapplications(wewillseesome)– Understandtheworld
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Remark
• Inmostofthegamesseeninthislecture,theactionsetswerefinite(i.e.,playershadafinitenumberofactionstochoosefrom)
• Thisisnotageneralthing:wewillseemanygameswithcontinuousactionsets(exercisesandnextlectures)– Example:companieschoosingprices
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