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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