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Chapter3RepresentationofGamesWearenowreadytoformallyintroducegamesandsomefundamentalconcepts,suchasastrategy. Inordertoanalyzeastrategicsituations,oneneedstoknow

whotheplayersare,whichactionsareavailabletothem,howmucheachplayervalueseachoutcome,whateachplayerknows.

Agame isjusta formal representation of the above information. This isusually doneinoneofthefollowingtwoways:

1. The extensive-form representation,in which the above information is explicitlydescribedusinggametreesandinformationsets;

2. Thenormal-form(orstrategic-form)representation, inwhichtheabove informa-tionissummarized byuseofstrategies.

Bothformsofrepresentationareuseful intheironway,andIwillusebothrepresenta-tionsextensivelythroughoutthecourse.

Itisimportanttoemphasizethat,whendescribingwhataplayerknows,oneneedstospecifynotonlywhatheknowsaboutexternalparameters,suchasthepayoffs,butalsowhatheknowsabouttheotherplayersknowledgeandbeliefsabouttheseparameters,

29

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30 CHAPTER3. REPRESENTATIONOFGAMESaswellaswhatheknowsabouttheotherplayersknowledgeofhisownbeliefs,andsoon. Inbothrepresentationssuchinformationisencodedinaneconomicalmanner. Inthefirsthalfofthiscourse,wewillfocusonnon-informationalissues,byconfiningourselvestothegamesofcompleteinformation,inwhicheverythingthatisknownbyaplayerisknownbyeverybody. Inthesecondhalf,wewillfocusoninformationalissues,allowingplayers to have asymmetric information, so that one may knowa piece of informationthatisnotknownbyanother.

Theoutlineofthislectureisasfollows. Thefirstsectionisdevotedtotheextensive-formrepresentationofgames. Thesecondsectionisdevotedtotheconceptofstrategy.The third section is devoted to the normal-form representation, and the equivalencebetweenthetworepresentations. Thefinalsectioncontainsexercisesandsomeoftheirsolutions.

3.1 Extensive-formRepresentationTheextensive-formrepresentationofagamecontainsalltheinformationaboutthegameexplicitly,bydefiningwhomoveswhen,whateachplayerknowswhenhemoves,whatmoves are available to him, and where each move leads to, etc. This is done by use of a gametree andinformationsetsaswellasmorebasicinformationsuchasplayersandthepayoffs.3.1.1 GameTreeDefinition3.1 Atreeisasetofnodesanddirectededgesconnectingthesenodessuchthat

1. thereisaninitialnode,forwhichthere isno incomingedge;2.foreveryothernode,there isexactlyone incomingedge;3.forany twonodes, there isauniquepaththatconnect thesetwonodes.Foravisualaid,imaginethebranchesofatreearisingfromitstrunk. Forexample,

thegraph inFigure3.1 isatree. There isauniquestartingnode,and itbranchesoutfrom there without forming a loop. It does look like a tree. On the other hand, the

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313.1. EXTENSIVE-FORMREPRESENTATIONgraphs in Figure 3.2 are not trees. In the graph on the left-hand side, there are twoalternativepathstonodeAfromtheinitialnode,onethroughnodeBandonethroughnodeC.Thisviolatesthethirdcondition. (Here,thesecondconditioninthedefinitionisalsoviolated,astherearetwoincomingedgestonodeA.)Ontheright-handside,thereis no path that connects the nodes x and y, once again violating the third condition.(Onceagain,thesecondconditionisalsoviolated.)

Figure3.1: Atree.

x

A

B

C y

Figure3.2: TwographsthatarenotatreeNote that edges (or arrows)come with labels, whichcan be same for two different

arrows. Inagametreetherearetwotypesofnodes,terminal nodes,atwhichthegameends,andnon-terminal nodes,atwhichaplayerwouldneedtomakeafurtherdecision.Thisisformallystatedasfollows.Definition3.2 The nodes that are notfollowed by another node are called terminal.

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333.1. EXTENSIVE-FORMREPRESENTATIONanallocationofnon-terminalnodesofthetreetotheplayers,an informational partition of the non-terminal nodes (to be made precise in the

nextsubsection),andpayoffsforeachplayerateachterminalnode.

Players Thesetofplayers consists of thedecisionmakers oractors whomakesomedecisionduringthecourseofthegame. SomegamesmayalsocontainaspecialplayerNature(orChance)thatrepresenttheuncertaintytheplayersface,asitwillbeexplainedinSubsection3.1.4. Thesetofgamesisoftendenotedby

={1 2 }and aredesignatedasgenericplayers.OutcomesandPayoffs Thesetofterminalnodesoftendenotedby. At a terminal node, the game has ended, leading to some outcome. At that point, one specifies apayoff,whichisarealnumber,foreachplayer. Themapping

:RthatmapseachterminalnodetothepayoffofplayeratthatnodeistheVon-NeumannandMorgensternutilityfunctionofplayer. Recallfromthepreviouschapterthatthismeans that player triestomaximizetheexpectedvalueof. That is, given any two lotteriesandon,hepreferstoifandonlyifleadstoahigherexpectedvaluefor

P Pfunction thandoes,i.e., () () ()(). Recallalsothatthese preferencesdonotchangeifwemultiplyallpayoffs with a fixedpositivenumberoraddafixednumbertoallpayoffs. Thepreferencesdochangeunderanyothertransformation.DecisionNodes Inanon-terminalnode,anewdecisionistobemade. Hence,inthedefinitionofagame,aplayerisassignedtoeachnon-terminalnode. This istheplayerwhowill make the decision at that point. Towardsdescribing thedecision problemofthe player at the time, one defines the available choices to the player at the moment.These are the outgoing arrows at the node, each of them leading to a different node.Each of these choices is also called move or action (interchangeably). Note that the

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353.1. EXTENSIVE-FORMREPRESENTATIONThemeaningofaninformationsetisthatwhentheindividualisinthatinformation

set, he knows that one of the nodes in the information set is reached, but he cannotrule out any of the nodes in the information set. Moreover, in a game, the information setbelongstotheplayerwho istomove inthegiven informationset, representinghisuncertainty. That is, the player who is to move at the information set is unable todistinguishbetweenthepointsintheinformationset,butabletodistinguishbetweenthepointsoutsidetheinformationsetfromthoseinit. Therefore,theabovedefinitionwouldbemeaninglesswithoutcondition1,whilecondition2requiresthattheplayerknowshisavailablechoices. Thelatterconditioncanbetakenasasimplifyingassumption. Ialsoreferto informationsetsashistory andwrite foragenerichistoryatwhichplayermoves.

Foranexample,considerthegame inFigure3.5. Here,Player2knowsthatPlayer1 has taken action orandnotaction;butPlayer2cannotknowforsurewhether

11 has taken or.1 x

T B

2

LR L

R

Figure3.5: Agame

Example3.2(MatchingPennieswithPerfectInformation) In Figure 3.4, theinformational partition is very simple. Every information set has only one element.Hence,there isnouncertaintyregardingthepreviousplay inthegame.

1 Throughoutthecourse,theinformationsetsaredepictedeitherbycircles(asinsets),orbydashedcurvesconnectingthenodesintheinformationsets,dependingonconvenience. Moreover,theinforma-tionsetswithonlyonenodeinthemaredepictedinthefigures. Forexample,inFigure3.5,theinitialnodeis inan informationsetthatcontainsonlythatnode.

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36 CHAPTER3. REPRESENTATIONOFGAMESA game is said to have perfect information if every information set has only one

element. Recall that in a tree, each node is reached through a unique path. Hence,inaperfect-informationgame,aplayercanconstructthepreviousplayperfectly. ForinstanceinFigure3.4,Player2knowswhetherPlayer1choseHeadorTail. AndPlayer1knowsthatwhenheplaysHeadorTail,Player2willknowwhatPlayer1hasplayed.3.1.4 NatureasaplayerandrepresentationofuncertaintyThe set ofplayers includes thedecisionmakers takingpart inthe game. However, inmanygamesthereisroomforchance,e.g. thethrowofdiceinbackgammonorthecarddraws in poker. More broadly, the players often face uncertainty about some relevantfact,includingwhattheotherplayersknow. Inthatcase,onceagainchanceplaysarole(as

a

representation).

To

represent

these

possibilities

we

introduce

afi

ctional

player:

Nature. Thereisnopayofffor Nature at end nodes, and every time a node is allocated toNature,aprobabilitydistributionoverthebranchesthatfollowneedstobespecified,e.g.,Tailwithprobabilityof1/2andHeadwithprobabilityof1/2. Notethatthisisthesameasaddinglotteriesintheprevioussectiontothegame.

Foranexample,considerthegameinFigure3.6. Inthisgame,afaircoinistossed,wheretheprobabilityofHeadis1/2. IfHeadcomesup,Player1choosesbetweenLeftandRight;ifTailcomesup,Player2choosesbetweenLeftandRight. Thepayoffs also dependonthecointoss.

(5, 0)Left

1

Head

1/2 Right (2, 2)

Nature(3, 3)

1/2 Left2Tail

Right (0, -5)

Figure3.6: Agamewithchance

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373.1. EXTENSIVE-FORMREPRESENTATION3.1.5 CommonlyKnownAssumptionsThe structure of agame is assumedtobeknownbyall the players, and it is assumedthatallplayersknowthestructureandsoon. That is, inamore formal language,thestructureofgameiscommonknowledge.2 Forexample,inthegameofFigure3.5,Player1knowsthatifhechooses or,Player2willknowthatPlayer1haschosenoneoftheaboveactionswithoutbeingabletoruleouteitherone. Moreover,Player2knowsthatPlayer 1 has the above knowledge, and Player 1 knows that Player 2 knows it, and soon. Using informationsetsandrichergametrees,onecanmodelarbitraryinformationstructures like this. For example, one could also model a situation in which Player 1doesnotknowwhetherPlayer2coulddistinguishtheactions and. One could do thatbyhavingthree informationset forPlayer2;oneofthem isreachedonlyafter,one of them is reached only after andoneofthemcanbereachedafterboth and.TowardsmodelinguncertaintyofPlayer1,onewouldfurtherintroduceachancemove,whoseoutcomeeitherleadstothefirsttwoinformationsets(observablecase)ortothelastinformationcase(unobservablecase).

Exercise3.1 WritethevariationofthegameinFigure3.5,inwhichPlayer1believesthatPlayer2candistinguishactions andwithprobability13andcannotdistinguishthemprobability23,andthisbeliefs iscommonknowledge.

To sum up: At any node, the following are known: which player is tomove, whichmoves are available to the player, and which information set contains the node, sum-marizingtheplayers informationatthenode. Ofcourse, iftwonodesare inthesameinformationset,theavailablemovesinthesenodesmustbethesame,forotherwisetheplayercoulddistinguishthenodesbytheavailablechoices. Again,alltheseareassumedtobecommonknowledge.

2 Formally, a proposition is said to be common knowledge if all of the following are true: istrue; everybody knows that is true; everybody knows that everybody knows that is true; . . . ; everybodyknowsthat... everybodyknowsthat istrue,adinfinitum.

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38 CHAPTER3. REPRESENTATIONOFGAMES3.2 StrategiesDefinition3.6 A strategy ofaplayer isacompletecontingent-plandeterminingwhichactionhewilltakeateachinformationsetheistomove(includingtheinformationsetsthatwillnotbereachedaccordingtothisstrategy).Moremathematically,astrategyofaplayerisafunction thatmapseveryinformationset ofplayertoanactionthatisavailableat.

Itisimportanttonotethefollowingthreesubtleties inthedefinition.1. One must assign a move to every information set of the player. (If we omit to

assign a move foran information set, wewouldnot knowwhat theplayerwouldhavedonewhenthat informationsetisreached.)

2. Theassignedmovemustbeavailableattheinformationset. (Iftheassignedmoveis not available at an information set, then the plan would not be feasible as itcouldnotbeexecutedwhenthatinformationsetisreached.)

3. At all nodes in a given information set, the player plays the same move. (Afterall,theplayercannotdistinguishthosenodesfromeachother.)

Example3.3(MatchingPennieswithPerfectInformation) InFigure3.4,Player1hasonlyoneinformationset. Hence,thesetofstrategiesforPlayer1is{Head,Tail}.On the other hand, Player 2 has two information sets. Hence, a strategy of Player 2determines what to do at each information set, i.e., depending on what Player 1 does.So,herstrategiesare:

=HeadifPlayer1playsHead,andHeadifPlayer1playsTail; =HeadifPlayer1playsHead,andTailifPlayer1playsTail; =TailifPlayer1playsHead,andHeadifPlayer1playsTail;

=

Tail

if

Player

1plays

Head,

and

Tail

if

Player

1plays

Tail.

Example3.4 InFigure3.5,bothplayershaveone informationset. Hence, thesetsofstrategiesforPlayers1and2are

1 ={ } and2 ={ }

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393.2. STRATEGIESrespectively. AlthoughPlayer2movesat twodifferentnodes, theyareboth inthesameinformationset. Hence,sheneedstoeitherplayatbothnodesoratbothnodes.

For certain purposes it might suffice to look at the reduced-form strategies. A re-duced

form

strategy

is

defi

nedas

an

incomplete

contingent

plan

that

determines

which

actiontheagentwilltakeateach informationsethe istomoveandthathasnotbeenprecludedbythisplan. Butformanyotherpurposesweneedtolookatallthestrategies.Throughoutthecourse,wemustconsiderallstrategies.

Whataretheoutcomesofstrategiesofplayers? Whatarethepayoffsgeneratedbythosestrategies? Towardsansweringthesequestions,weneedfirstacoupleofjargon.Definition3.7 In a game with players ={1 }, a strategyprofileisa list

= (1

)

ofstrategies,oneforeachplayer.Definition3.8 In a game without Nature, each strategy profile leads to a unique terminal node (), called the outcome of . The payoff vectorfrom strategy is thepayoffvectorat ().

Sometimestheoutcomeisalsodescribedbytheresultinghistory,whichcanalsobecalledasthepathofplay.Example3.5(MatchingPennieswithPerfectInformation) InFigure3.4,ifPlayer1playsHeadandPlayer2plays, thentheoutcome is

both players choose Head,and the payoff vector is (1 1). If Player 1 plays Head and Player 2 plays , the outcome is the same, yielding the payoff vector (1 1). If Player 1 plays Tail andPlayer2playsHT, thentheoutcome isnow

both players choose Tail,butthepayoffvector is(1 1)onceagain. Finally, ifPlayer1playsTailandPlayer2plays, thentheoutcome is

Player 1 chooses Tail and Player 2 chooses Head,

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40 CHAPTER3. REPRESENTATIONOFGAMESand the payoff vector is (11). One can compute the payoffsfor the other strategyprofilessimilarly.

IngameswithNature,astrategyprofileleadstoaprobabilitydistribution onthesetofterminalnodes. Theoutcomeofthestrategyprofileisthentheresultingprobabilitydistribution. The payoffvector fromthe strategy profile is the expected payoffvectorundertheresultingprobabilitydistribution.Example3.6(AgamewithChance) InFigure3.5,eachplayerhastwostrategies,LeftandRight. Theoutcomeofthestrategyprofile(Left,Left) isthe lotterythat

Nature chooses Head and Player 1 plays Leftwithprobability1/2and

Nature chooses Tail and Player 2 plays Leftwithprobability1/2. Hence,theexpectedpayoffvector is

1 1 () = (5 0) + (3 3)=(4 32)

2 2Sometimes,itsufficestosummarizealloftheinformationabovebythesetofstrate-

gies and the utility vectors from the strategy profiles, computed as above. Such asummaryrepresentationiscalledformal-formorstrategic-formrepresentation.

3.3 Normal formDefinition3.9 (Normalform)Agame isany list

= (1 ;1 )where,for each ={1 }, is the set of all strategies that are available toplayer, and

:1 RisplayersvonNeumann-Morgensternutilityfunction.

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413.3. NORMALFORMNoticethataplayersutilitydependsnotonlyonhisownstrategybutalsoonthe

strategiesplayedbyotherplayers. Moreover,eachplayertriestomaximizetheexpectedvalue of (wheretheexpectedvaluesarecomputedwithrespecttohisownbeliefs);inotherwords, isavonNeumann-Morgensternutilityfunction. Wewillsaythatplayerisrational iffhetriestomaximizetheexpectedvalueof (givenhisbeliefs).

Itisalsoassumedthatitiscommonknowledgethattheplayersare={1 },thatthesetofstrategiesavailabletoeachplayeris , and that each triestomaximizeexpectedvalueof givenhisbeliefs.

When there are only two players, we can represent the normal form game by a bimatrix(i.e.,bytwomatrices):

1\2

0

2

11

41 32Here, Player 1 has strategies and , and Player 2 has the strategies and. In each box the first number is Player 1s payoffandthesecondoneisPlayer2spayoff(e.g.,1() = 0,2() = 2.)3.3.1 FromExtensiveFormtoNormalFormAsithasbeendescribedindetailintheprevioussection,inanextensiveformgame,thesetofstrategiesisthesetofallcompletecontingentplans,mappinginformationsetstotheavailablemoves. Moreover,eachstrategyprofile leadstoanoutcome(), which is ingeneralprobabilitydistributiononthesetofterminalnodes. Thepayoffvector istheexpectedpayoffvector from(). Onecanalwaysconvertanextensive-formgametoanormalformgameinthisway.Example3.7(MatchingPennieswithPerfectInformation) InFigure3.4,basedon the earlier analyses, the normal or the strategicform game corresponding to thematchingpennygamewithperfect informationis

11 11 11 1111 11 11 11

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42 CHAPTER3. REPRESENTATIONOFGAMES1

2

HeadTail

head tail head tail

(-1,1) (1,-1) (1,-1) (-1,1)

Figure3.7: MatchingPenniesGameInformation sets are very important. To see this, consider the following standard

matching-pennygame. Thisgamehasimperfectinformation.

Example3.8(MatchingPennies) Consider the game in Figure 3.7. This is thestandard matching penny game, which has imperfect information as the players movesimultaneously. In this game, each player has only two strategies: Head and Tail. Thenormal-formrepresentationis

1\2 head tail Head 11 11Tail 11 11

The two matching penny games may appear similar (in extensive form), but theycorrespondtotwodistinctsituations. Underperfect informationPlayer2knowswhatPlayer1hasdone,whilenobodyknowsabouttheotherplayersmoveundertheversionwithimperfectinformation.

As mentioned above, when there are chance moves, one needs to compute the ex-pectedpayoffs inordertoobtainthenormal-formrepresentation. This is illustratedinthenextexample.

Example3.9(AgamewithNature) Asmentioned, inFigure3.6, eachplayerhastwo strategies, Left and Right. Following the earlier calculations, the normal-formrep-

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433.3. NORMALFORM2

1HH

HT THTT

H T1

H T

1 1

1H T

1 1

1H T

1 11 1

1 1 1 11 1 1 1

Figure3.8: Amatchingpennygamewithperfectinformation?resentationisobtainedasfollows:

1\2 Left Right Left 432 5252

Right 5252 132The payofffrom (Left,Left) has been computed already. The payofffrom (Left, Right)computed as

1 1(50) + (05)=(5252)

2 2Whilethereisauniquenormal-formrepresentationforanyextensive-formgame(up

to a relabeling of strategies), there can be many extensive-form games with the samenormal-form representation. After all, any normal-form game can also be representedasasimultaneousactiongame inextensive form. Forexample, thenormal-formgameofmatchingpennieswithperfectinformationcanalsoberepresentedasinFigure3.8.3.3.2 MixedStrategiesIn many cases a player may not be able to guess exactly which strategies the otherplayersplay. Inordertocoverthesesituationsweintroducethemixedstrategies:Definition3.10 Amixedstrategyofaplayer isaprobabilitydistributionoverthesetofhisstrategies.

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44 CHAPTER3. REPRESENTATIONOFGAMESIfplayerhasstrategies ={1 2 }, then a mixed strategy forplayer

isafunctionon suchthat0 () 1 and

(1

) + (2

) + + () = 1

Therearemanyinterpretationsformixedstrategies,fromdeliberaterandomization(asincointossing)toheterogeneityof strategies inthe population. Inall cases, however,theyserveasadevicetorepresenttheuncertaintytheotherplayers faceregardingthestrategyplayedbyplayer. Throughoutthecourse, isinterpretedastheotherplayersbeliefs about the strategy player plays.

3.4 ExerciseswithSolutions1. Whatisthenormal-formrepresentationforthegameinFigure3.12?

Solution: Player1hastwoinformationsetswithtwoactionineach. Sincethesetofstrategiesisfunctionsthatmapinformationsetstotheavailablemoves,hehasthe

following

four

strategies:

.

The

meaning

here is straightforward:

assignstothefirstinformationsetandto the last information set. On the otherhand,Player2hasonlytwostrategies: and. Fillinginthepayoffs from thetree,oneobtainsthefollowingnormal-formrepresentation:

1\2

15 5 23 3 5 24 4

4 4

4 4 4 4

2. [Midterm1,2001]Findthenormal-formrepresentationofthegameinFigure3.9.

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453.4. EXERCISESWITHSOLUTIONS

Figure3.9:Solution:

1\2

21 21 2121 21 2121 21 2121 21 2112 31 1312 31 3112 13 1312 13 31

3. [MakeupforMidterm1,2007]WritethegameinFigure3.10innormalform.Solution: TheimportantpointinthisexerciseisthatPlayer2hastoplaythesamemoveinagiveninformationset. Forexample,shecannotplayontheleftnodeandontherightnodeofhersecondinformationset. Hence,hersetofstrategiesis{}.

33 33 00 0033 33 00 0000 00 33 3300 00 33 3322 22 11 1322 22 11 11

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46 CHAPTER3. REPRESENTATIONOFGAMES1

A cB

2b

a b a b a1

23 0 0 323 0 0 3

2

X Y

x y x y1 1 1 11 3 1 1

Figure3.10:4. [Make up for Midterm 1, 2007] Write the following game in normal form, where

thefirstentryisthepayoffofstudentandthesecondentryisthepayoffofProf.healthy

sick.5

.5student

regularMake up

student

regularMake up

Prof

same new same new

2

1 0

-1

4

-1

1

-c4

01

-c

healthysick

.5.5

student

regularMake up

student

regularMake up

Prof

same new same new

2

1 0

-1

4

-1

1

-c4

01

-c

Solution: Writethestrategiesofthestudentas,,, and , where meansRegularwhenHealthyandMakeupwhenSick, meansMakeup

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3.5. EXERCISES 47

Figure3.11:when Healthy and Regular when Sick, etc. The normal form game is as follows:

Student\Prof same new

10 10312 32(1)221 12(1+)2

412 1Here, the payoffs are obtained by taking expectations over whether the studentis healthy or sick. For example, leads to (21) and (01) with equal prob-abilities, yielding (10), regardless of the strategy of Prof. On the other hand,(new)leadsto(21)and(1)withequalprobabilities,yielding(32(1)2).

5. [Midterm2006]WritethegameinFigure3.11innormalform.Solution:

1\2

11 11 11 11 11 11 11 1132 32 32 32 31 31 00 0000 00 00 00 01 23 01 23

3.5 Exercises1. [Midterm1,2010]WritethegameinFigure3.13innormalform.

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48 CHAPTER3. REPRESENTATIONOFGAMES

1

D

(4,4)

A 2

(5,2)

ad

(3,3)

1(1,-5)

Figure3.12: Acentepede-likegame

Figure3.13:

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493.5. EXERCISES2. [Midterm1,2005]WritethegameinFigure3.14innormalform.3. [Midterm1,2004]WritethegameinFigure3.15innormalform.

Figure3.14:4. [Midterm1,2007]Writethefollowinggameinnormalform.

Figure3.15:

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50 CHAPTER3. REPRESENTATIONOFGAMES

Figure3.16:5. [Homework 1, 2011] Find the normal-form representation of the extensive-form

gameinFigure3.16.

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