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ii“Alles”—2014/5/8—11:19—pagei—#1iiiiiiGameTheoryThroughExamples i i “Alles”—2014/5/8—11:36—pageii—#2 i i i i i 2014bytheMathematicalAssociationofAmerica,Inc.ElectroniceditionISBN978-1-61444-115-1 ii“Alles”—2014/5/8—11:19—pageiii—#3iiiiiiGameTheoryThroughExamplesErichPrisnerFranklinUniversitySwitzerlandPublishedandDistributedbyTheMathematicalAssociationofAmerica ii“Alles”—2014/5/8—11:19—pageiv—#4iiiiiiCouncilonPublicationsandCommunicationsFrankFarris,ChairCommitteeonBooksFernandoGouvˆea,ChairClassroomResourceMaterialsEditorialBoardSusanGStaples,EditorMichaelBardzellJenniferBergnerCarenLDiefenderferChristopherHallstromCynthiaJHuffmanPaulRKlingsbergBrianLinsMaryMorleyPhilipPMummertBarbaraEReynoldsDarrylYong ii“Alles”—2014/5/8—11:19—pagev—#5iiiiiiCLASSROOMRESOURCEMATERIALSClassroomResourceMaterialsisintendedtoprovidesupplementaryclassroommaterialforstudents—laboratoryexercises,projects,historicalinformation,textbookswithunusualapproachesforpresentingmath-ematicalideas,careerinformation,etc.101CareersinMathematics,3rdeditioneditedbyAndrewSterrettArchimedes:WhatDidHeDoBesidesCryEureka?,ShermanSteinCalculus:AnActiveApproachwithProjects,StephenHilbert,DianeDriscollSchwartz,StanSeltzer,JohnMaceli,andEricRobinsonCalculusMysteriesandThrillers,R.GrantWoodsConjectureandProof,Mikl´osLaczkovichCounterexamplesinCalculus,SergiyKlymchukCreativeMathematics,H.S.WallEnvironmentalMathematicsintheClassroom,editedbyB.A.FusaroandP.C.KenschaftExcursionsinClassicalAnalysis:PathwaystoAdvancedProblemSolvingandUndergraduateResearch,byHongweiChenExplorationsinComplexAnalysis,MichaelA.Brilleslyper,MichaelJ.Dorff,JaneM.McDougall,JamesS.Rolf,LisbethE.Schaubroeck,RichardL.Stankewitz,andKennethStephensonExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.WellerExploringAdvancedEuclideanGeometrywithGeoGebra,GerardA.VenemaGameTheoryThroughExamples,ErichPrisnerGeometryFromAfrica:MathematicalandEducationalExplorations,PaulusGerdesHistoricalModulesfortheTeachingandLearningofMathematics(CD),editedbyVictorKatzandKarenDeeMichalowiczIdenticationNumbersandCheckDigitSchemes,JosephKirtlandInterdisciplinaryLivelyApplicationProjects,editedbyChrisArneyInverseProblems:ActivitiesforUndergraduates,CharlesW.GroetschKeepingitR.E.A.L.:ResearchExperiencesforAllLearners,CarlaD.MartinandAnthonyTongenLaboratoryExperiencesinGroupTheory,EllenMaycockParkerLearnfromtheMasters,FrankSwetz,JohnFauvel,OttoBekken,BengtJohansson,andVictorKatzMathMadeVisual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaandRogerB.NelsenMathematicsGalore!:TheFirstFiveYearsoftheSt.MarksInstituteofMathematics,JamesTantonMethodsforEuclideanGeometry,OwenByer,FelixLazebnik,andDeirdreL.SmeltzerOrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.S´anchezOvalTrackandOtherPermutationPuzzles,JohnO.KiltinenParadoxesandSophismsinCalculus,SergiyKlymchukandSusanStaplesAPrimerofAbstractMathematics,RobertB.AshProofsWithoutWords,RogerB.NelsenProofsWithoutWordsII,RogerB.NelsenRediscoveringMathematics:YouDotheMath,ShaiSimonson ii“Alles”—2014/5/8—11:19—pagevi—#6iiiiiiSheDoesMath!,editedbyMarlaParkerSolveThis:MathActivitiesforStudentsandClubs,JamesS.TantonStudentManualforMathematicsforBusinessDecisionsPart1:ProbabilityandSimulation,DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklicStudentManualforMathematicsforBusinessDecisionsPart2:CalculusandOptimization,DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklicTeachingStatisticsUsingBaseball,JimAlbertVisualGroupTheory,NathanC.CarterWhichNumbersareReal?,MichaelHenleWritingProjectsforMathematicsCourses:CrushedClowns,Cars,andCoffeetoGo,AnnalisaCrannell,GavinLaRose,ThomasRatliff,andElynRykkenMAAServiceCenterP.O.Box91112Washington,DC20090-11121-800-331-1MAAFAX:1-301-206-9789 ii“Alles”—2014/5/8—11:19—pagevii—#7iiiiiiContentsPrefacexvi1Theory1:Introduction11.1What'saGame?..........................................11.2Game,Play,Move:SomeDenitions..............................11.3ClassicationofGames......................................2Exercises................................................32Theory2:SimultaneousGames42.1NormalForm—BimatrixDescription...............................42.1.1TwoPlayers........................................42.1.2TwoPlayers,Zero-sum..................................52.1.3ThreeorMorePlayers..................................52.1.4SymmetricGames....................................62.2WhichOptiontoChoose.....................................62.2.1MaximinMoveandSecurityLevel............................62.2.2DominatedMoves....................................72.2.3BestResponse......................................82.2.4NashEquilibria......................................92.3AdditionalTopics.........................................132.3.1BestResponseDigraphs.................................132.3.22-PlayerZero-sumSymmetricGames..........................14Exercises................................................15Project1:Reactingfastorslow.....................................183Example:SelectingaClass193.1ThreePlayers,TwoClasses....................................193.1.1“Ilikeyouboth”.....................................193.1.2DislikingtheRival....................................213.1.3Outsider..........................................213.2LargerCases...........................................223.3Assumptions...........................................23Exercises................................................23Project2.................................................23Project3.................................................24Project4.................................................24vii ii“Alles”—2014/5/8—11:19—pageviii—#8iiiiiiviiiContents4Example:DoctorLocationGames254.1DoctorLocation..........................................254.1.1AnExampleGraph....................................264.1.2No(Pure)NashEquilibrium?..............................274.1.3HowGoodaretheNashEquilibriaforthePublic?...................284.2Trees................................................284.3MorethanoneOfce(optional).................................31Exercises................................................31Project5:DoctorlocationonMOPs..................................33Project6.................................................33Project7.................................................335Example:RestaurantLocationGames345.1AFirstGraph...........................................355.2ASecondGraph..........................................365.3ExistenceofPureNashEquilibria................................375.4MorethanoneRestaurant(optional)...............................38Exercises................................................396UsingExcel426.1SpreadsheetProgramslikeExcel.................................426.2Two-PersonSimultaneousGames................................436.3Three-PersonSimultaneousGames................................43Exercises................................................43Project8:SimultaneousQuatro-Uno..................................44Project9:RestaurantLocationGames.................................44Project10:5Knights..........................................45Project11:5Cardinals.........................................457Example:ElectionI477.1FirstExample...........................................477.2SecondExample.........................................487.3TheGeneralModel........................................507.4ThirdExample..........................................507.5TheEightCases..........................................517.6VotingPowerIndices(optional).................................51Exercises................................................528Theory3:SequentialGamesI:PerfectInformationandnoRandomness538.1ExtensiveForm:GameTreeandGameDigraph.........................538.2AnalyzingtheGame:BackwardInduction............................568.2.1FiniteGames.......................................568.2.2TheProcedure......................................578.2.3Zermelo'sTheorem....................................598.3AdditionalTopics.........................................598.3.1RealityCheck.......................................598.3.2PlayingitSafe—GuaranteedPayoffs..........................618.3.3Two-personZero-sumGames..............................63 ii“Alles”—2014/5/8—11:19—pageix—#9iiiiiiContentsix8.3.4BreakingTies.......................................648.3.5ExistingGames......................................648.3.6GreedyStrategies.....................................65Exercises................................................65Project12:TAKESOME........................................69Project13:WHO'sNEXT(n)......................................69Project14:LISA'sGAME.......................................69Project15:2-AUCTION........................................69Project16:3-AUCTION........................................699Example:DividingAFewItemsI709.1GreedyStrategy..........................................709.2BackwardInduction........................................719.2.1GameTree........................................719.2.2GameDigraph......................................719.2.3Example:GameDigraphforABBAB..........................729.3AnAbbreviatedAnalysis.....................................729.3.1WhyitMatters:Complexity(optional).........................749.4Bottom-UpAnalysis.......................................759.5InterdependenciesbetweentheItems(optional).........................76Exercises................................................7610Example:ShubikAuctionI77Exercises................................................79Project17:SHUBIKAUCTION....................................7911Example:SequentialDoctorandRestaurantLocation8011.1GeneralObservationsforSymmetricGames...........................8011.2DoctorLocation..........................................8111.3Constant-SumGames.......................................8211.4RestaurantLocation........................................8311.5NashEquilibriaandFirstMoverAdvantageforSymmetricGames...............84Exercises................................................84Project18................................................85Project19:HostileversusFriendlyPlay................................8512Theory4:Probability8612.1Terminology............................................8612.2ComputingProbabilities.....................................8812.2.1EquallyLikelySimpleEvents..............................8812.2.2SimpleEventsnotEquallyLikely............................8812.3ExpectedValue..........................................8912.4MultistepExperiments......................................9112.4.1ProbabilityTrees.....................................9112.4.2ConditionalProbabilities.................................9112.4.3ProbabilityDigraphs...................................9312.5RandomnessinSimultaneousGames...............................9412.6CountingwithoutCounting....................................95 ii“Alles”—2014/5/8—11:19—pagex—#10iiiiiixContentsExercises................................................95Project20:Tennis............................................97Project21:FinalExam.........................................9813France165499Exercises................................................10114Example:DMASoccerI10214.11-Round2-StepExperimentforGivenPlayerDistributions...................10314.2ExpectedGoalDifferencefortheOne-RoundGame.......................10414.33-RoundsExperimentforGivenPlayerDistributions......................10514.4StaticThree-roundGame.....................................10714.5StaticNine-roundDMASoccer..................................108Exercises................................................108Project22:DMA6*Soccer.......................................109Project23:DMA7*Soccer.......................................10915Example:DividingAFewItemsII11015.1GoalsofFairnessandEfciency.................................11015.1.1Fairness..........................................11015.1.2Efciency.........................................11115.1.3ThreeAdditionalFeatures................................11115.1.4MechanismDesign....................................11215.2SomeGames...........................................11215.2.1SelectingonebyoneGames...............................11215.2.2CutandChoose......................................11215.2.3RandomandExchange..................................11315.3Examples.............................................11315.4ComparisonoftheGamesforSevenItemsandCompleteInformation.............11515.4.1OpposingorSimilarPreferences.............................11615.5IncompleteInformation......................................118Exercises................................................119Project24:DividingveitemsA....................................120Project25:DividingveitemsB....................................12016Theory5:SequentialGameswithRandomness12116.1ExtensiveFormExtended.....................................12116.2AnalyzingtheGame:BackwardInductionagain.........................12116.3DecisionTheory:AloneagainstNature.............................122Exercises................................................125Project26:JobInterviews.......................................127Project27:5Envelopes.........................................127Project28:Oh-NoorOh-No6.....................................128Project29:34versionofPolyominoRECTHESQUAREwithrandomness............12817Example:SequentialQuizShowI12917.1CandidateswithLittleKnowledge................................12917.1.1MoreMaybeLess....................................130 ii“Alles”—2014/5/8—11:19—pagexi—#11iiiiiiContentsxi17.2OneCandidateKnowsMore...................................13117.2.1CindyKnowsoneAnswertobeFalse..........................133Exercises................................................133Project30:SEQUENTIALQUIZSHOW,cleverAnn.........................134Project31:SEQUENTIALQUIZSHOW,cleverBeth.........................13418LasVegas1962135Exercises................................................13819Example:MiniBlackjackandCardCounting13919.1TheBasicGame..........................................13919.2PlayingagainsttheHouse.....................................14219.2.1HowLikelyaretheDistributions?............................14319.2.2BettingHighandLow..................................14519.2.3Reshufing........................................146Exercises................................................147Project32:8ROUNDSBLACKORWHITE.............................147Project33:xROUNDSRED,GREEN,ORBLUE..........................148Project34:MINIBLACKJACK....................................14820Example:Duel14920.1OneBullet.............................................14920.1.1AnalysisofOne-bulletVariantswithIncreasingProbabilitieswithoutComputerHelp.15020.1.2AnalysisofDUEL(1;1jm;2m;3m;:::).........................15120.2TwoormoreBullets.......................................15220.2.1AfewCasesofDUEL(2;2jm;2m;3m;:::).......................153Exercises................................................154Project35:DrunkAdam........................................154Project36:Howmoredangerousweaponsaffectthestatebudgetandthehealthofcitizens.....154Project37:Selectingmbetween0.04and0.13............................154Project38:Whatduelsarebestforsociety?..............................15521SantaMonicainthe50s15622Theory6:ExtensiveFormofGeneralGames15922.1ExtensiveFormandInformationSets...............................15922.2NoBackwardInductionforImperfectInformation........................16322.3Subgames.............................................16422.4Multi-roundGames........................................16522.5WhyTreesforImperfectInformation?..............................165Exercises................................................16623Example:ShubikAuctionII16923.1PossibleSuddenEnd.......................................16923.2ImperfectandIncompleteInformation..............................17223.3TheAuctioneerEnterstheGame(optional)...........................172Exercises................................................174Project39................................................174 ii“Alles”—2014/5/8—11:19—pagexii—#12iiiiiixiiContentsProject40................................................174Project41:SHUBIKAUCTION(45;35;6;p).............................175Project42:SHUBIKAUCTION(A;B;C;n;p)............................17524Theory7:NormalFormandStrategies17624.1PureStrategies..........................................17624.1.1ReducedPureStrategies.................................17724.2NormalForm...........................................17724.3UsingToolsfromSimultaneousGamesfortheNormalForm..................18024.4SubgamePerfectness.......................................18024.5SpecialCaseofSequentialGameswithPerfectInformation...................182Exercises................................................18225Example:VNMPOKERandKUHNPOKER18525.1Description............................................18525.2VNMPOKER...........................................18725.3KUHNPOKER..........................................190Exercises................................................19126Example:WaitingforMr.Perfect19326.1TheLastRound..........................................19326.2TheEightPureStrategies.....................................19426.3ComputingthePayoffs......................................19426.4Domination............................................19526.5TheReducedNormalFormsintheThreeCases.........................19626.5.1TheCasep2C2p31.................................19626.5.2TheCasep2C2p3.21;Ô°1.................................19726.5.3TheCasep2C2p3D1.................................197Project43................................................198Project44................................................198Project45................................................198Project46................................................198Project47................................................198Project48................................................19827Theory8:MixedStrategies19927.1MixedStrategies.........................................19927.1.1BestResponse......................................20027.1.2Brown'sFictitiousPlay..................................20027.1.3MixedMaximinStrategy,MixedSecurityLevel,andLinearPrograms.........20227.2MixedNashEquilibria......................................20327.2.1Two-playerZero-sumGames...............................20327.2.2Non-Zero-sumGames..................................20327.3ComputingMixedNashEquilibria................................20427.3.1SmallTwo-playerZero-sumGames(optional).....................20527.3.2SolvingSmallnonZero-sumTwo-playerGamesbySolvingEquations(optional)...207Exercises................................................209Project49:Balanced3-spinnerduelwithveoptions.........................210 ii“Alles”—2014/5/8—11:19—pagexiii—#13iiiiiiContentsxiiiProject50:Balanced3-spinnerduel..................................210Project51:COLONELBLOTTO(4,9,9)...............................210Project52:IteratedCOLONELBLOTTO...............................211Project53:SimultaneousQuatro-Uno.................................211Project54................................................211Project55:4-roundWaitingforMr.Perfect..............................21128Princetonin195021229Example:AirportShuttle21529.1TheSimpleModel........................................21529.1.1TotheAirport.......................................21529.1.2FromtheAirport.....................................21829.1.3CombiningBoth.....................................21829.2ImpatientCustomers.......................................219Exercises................................................21930Example:ElectionII22030.1LeftOverfromElectionI.....................................22030.2MoreEffortintoLargeDistricts.................................22130.3DefendWhereAheadorAttackWhereWeak?..........................22230.4IsLargerBetter?.........................................22330.5ELECTION(7;8;13j1;1;2jx;x/...............................223Exercises................................................22431Example:VNMPOKER(2;r;m;n)22531.1TheCasenm21r.......................................22631.2BestResponses..........................................22731.3ReadingtheOpponent(optional).................................22731.4MixedNashEquilibriumfornm21r.............................22831.5SmallChangesintheParameters.................................229Exercises................................................23032Theory9:BehavioralStrategies23132.1BehavioralversusMixedStrategies................................23232.1.1CalculatingMixedStrategiesfromBehavioralStrategies................23332.1.2CalculatingBehavioralStrategiesfromMixedStrategiesforaGameTreewithPerfectRecall...........................................23332.1.3Kuhn'sTheorem.....................................235Exercises................................................23533Example:Multiple-RoundChicken23733.1OrdinaryChicken.........................................23733.2Two-roundChicken........................................23833.2.1GeneralizedBackwardInduction,usingtheExtensiveForm..............23833.2.2WorkingwiththeNormalForm.............................24033.2.3ConnectionsbetweenthetwoApproaches........................24033.3Three-roundChicken.......................................241 ii“Alles”—2014/5/8—11:19—pagexiv—#14iiiiiixivContentsExercises................................................243Project56................................................243Project57................................................243Project58................................................24334Example:DMASoccerII24434.1Multi-roundSimultaneousGames................................24434.2InformationSetsandMoves...................................24534.3TheOptimalThirdMoveinSelectedCases...........................24634.3.1ADetailedExample:(2,2)versus(3,1).........................24634.3.2ASecondExample:(1,3)versus(2,2).........................24934.4TheOptimalSecondMoveforSevenPositions.........................24934.5Couldn'tWeAnalyzetheWholeGame?.............................25134.6HowGoodaModelisit?.....................................25135Example:SequentialQuizShowII25235.1FixedCoalitions..........................................25235.1.1AnnandCindyFormaCoalition.............................25235.1.2AnnandBethFormaCoalition.............................25435.1.3BethandCindyFormaCoalition............................25435.2WhichCoalitionWillForm?...................................25435.2.1Fixed50:50Split.....................................25435.3AnotherVariant:SplitcanbeNegotiated.............................25535.4TheGrandCoalition.......................................25635.4.1TheCore.........................................25635.4.2TheShapleyValue....................................256Exercises................................................257Project59................................................25836Example:VNMPOKER(4,4,3,5)25936.1MixedNashEquilibria......................................25936.2PerformanceofPureStrategiesagainsttheMixedNashEquilibria...............26137Example:KUHNPOKER(3,4,2,3)26437.1FromBehavioralStrategiestoMixedStrategiestoExpectations................26537.2FromMixedStrategiestoBehavioralStrategies.........................266Exercises................................................26638Example:End-of-SemesterPokerTournament26838.1Expectations............................................26938.2Odds................................................27038.2.1ManyRounds.......................................27338.3TheFavoriteinKnockoutTournaments.............................27338.4SimilarityoftheDNA(optional).................................27438.5HowtoCreateyourownTournament...............................274Exercises................................................275Project60................................................275Project61................................................276 ii“Alles”—2014/5/8—11:19—pagexv—#15iiiiiiContentsxvProject62................................................27639Stockholm1994277Bibliography280Index284 ii“Alles”—2014/5/8—11:19—pagexvi—#16iiiiiiPrefaceWelcometogametheory,themathematicaltheoryofhowtoanalyzegamesandhowtoplaythemoptimally.Although“game”usuallyimpliesfunandleisure,gametheoryisaseriousbranchofmathematics.Gameslikeblackjack,poker,andchessareobviousexamples,buttherearemanyothersituationsthatcanbeformulatedasgames.Wheneverrationalpeoplemustmakedecisionswithinaframeworkofstrictandknownrules,andwhereeachplayergetsapayoffbasedonthedecisionsofalltheplayers,wehaveagame.Examplesincludeauctions,negotiationsbetweencountries,andmilitarytactics.Thetheorywasinitiatedbymathematiciansinthersthalfofthelastcentury,butsincethenmuchresearchingametheoryhasbeendoneoutsideofmathematics.Thisbookgivesanintroductiontogametheory,onanelementarylevel.Itdoesnotcoverallareasofaeldthathasburgeonedinthelastsixtyyears.Ittriestoexplainthebasicsthoroughly,sothatitisunderstandableforundergraduatesevenoutsideofmathematics,Ihopeinanentertainingway.Thebookdiffersfromothertextsbecauseitemphasizesexamples.Thetheoryisexplainedinninetheorychapters,andhowitisappliedisillustratedintwenty-fourexamplechapters,whereexamplesareanalyzedindetail.Eachexamplechapterusestoolsdevelopedintheorychapters.AudienceThetextcanbeusedfordifferentpurposesandaudiences.Itcanbeusedbystudentsofeconomics,politicalscience,orcomputerscience.Undergraduatemathematicsstudentsshouldenjoythebookaswellandprotfromitsapproach.Thebookmayalsobeusedasasecondarytext,orforindependentstudy,asitsmanyconcreteexamplescomplementexpositionsofthetheory.Ithinkthereisanotheraudiencethatcouldprotfromthebook,theoneIhadinmindwhenIwrotethebook,whichisundergraduateswhotakeamathematicsclasstofulllgeneraleducationorquantitativereasoningrequirements.ThetextbeganasanonlinetextforarstyearseminaratFranklinCollegein2007,whenIcouldn'tndanappropriatetext.Sincethen,IhaveusedversionsofthetextinourrstyearseminarandinaliberalartscourseIhavecalledIntroductiontoGameTheory.Gametheoryandmyapproachtoitarewellsuitedforthesecoursesbecause:Theunderlyingmathematicsisbasic:forexample,ndingminimaandmaxima,computingweightedaveragesandworkingsystematicallythroughtreesordigraphs.Themathematicaltopicsthatarecovered—probability,trees,digraphs,matrices,andalgorithms—areamongthemostusefultoolsusedinapplications.Gametheoryisappliedmathematics.Somewouldclaimitisnotevenmathematics,butpartofeco-nomics.The“whyarewedoingthis?”questionhasananobviousanswer.Studentsacceptthefactthatmathematicaltoolshavetobedevelopedforstudyingandanalyzinggames.Gametheorygivesopportunitiesforstudentstodoprojects.Gametheoryallowsaplayfulapproach.xvi ii“Alles”—2014/5/8—11:19—pagexvii—#17iiiiiiPrefacexviiFeaturesofthebookTherearethirty-eightchaptersofthreetypes:Ninechapterspresentthebasictheory.Theyarelistedastheorychaptersinthetableofcontents.Inmostcoursesallthesechapterswouldbecovered.ThereisanadditionalchapterexplaininghowtousetheExcelles,whichisalsorequiredreading.Fivechaptersdescribenotablelandmarksinthehistoryanddevelopmentofgametheory:the1654lettersbetweenPascalandFermat,whichmarkthebeginningofprobabilitytheory;PrincetonUniversitybefore1950,wheregametheorystartedasanacademicdiscipline;RANDCorporationinthe1950sandtheearlyoptimismthereaboutwhatgametheorycouldachieve;casinogames;andtheNobelprizesawardedtogametheorists.Thechaptersprovidebackgroundaboutsomeoftheimportantpersonsinthedevelopmentofgametheory,anddiscussgametheory'sroleinsociety.Thechaptersmightbeforreadingathomeforstudents.Theycouldalsobevehiclesforclassdiscussions.Therealcoreofthemanuscriptarethetwenty-fourchaptersthattreatconcreteexamples.Theydis-tinguishthebookfromothers.Thechaptersusethetheorydevelopedinthetheorychapters,andtheyusuallybuildonatmostoneotherexamplechapter,soaninstructorcanselectthosethatseemmostap-propriate.Inmyclasses,Iusuallydiscussabouteightexamples.Someoftheexamplechaptersprovideglimpsesintoareasthatarenotcoveredinthetheorychapters.ExamplesofthisaretreesinChapter4,votingpowerindicesinChapter7,complexityandbinomialcoefcientsinChapter9,statistics,mech-anismdesign,andincompleteinformationinChapter15,incompleteversusimperfectinformationinChapter23,workwithparametersandalgebrainChapter31,andcooperativegamesinChapter35.Therearetwoothermainfeaturesofthebook:theExcelspreadsheetsandJavascriptapplets.TheJavascriptappletsaresmallgames,inwhichstudentscantryoutmanyofthegamesdiscussedandanalyzedintheexamples.Studentscanplayagainstthecomputeroragainststudents.Beforereadingtheanalysisstudentsshouldhaveplayedthegamesanddevelopedsomeideasonhowtoplaythegame.TheJavascriptsarealsousedtoconrmtheoreticalresults.TheExcelspreadsheetseitheraregenerallyusableforanysimultaneous2-playergameorforevery2-playergameinnormalform,ortheyaredesignedforspecicgames.Theygivethestudentstoolstocheckwhatisdescribedinthetextwithouthavingtodomanyroutineortediouscalculations,andalsotoolstoapply“whatif”analysistogames.Studentswillsee,sometimesinunexpectedways,howchangingparametersmaychangestrategies,outcomes,andpayoffs.InmyopinionExcelisthebesttoolforthegametheoryinthisbook,betterthanMathematicaorMapleorMathCad.Sinceallstepsareelementary(nofunctionsareusedexcept,“Min”,“Max”,“Sum”,and“If”'),everythingcouldalsobedonebythestudentonasheetofpaper,atleastinprinciple.EverystudentshouldlearnsomeExcelanyway,anotherfeaturewhichmakesacoursebasedonthisbookverywell-suitedforgeneraleducationcorerequirements.FortheInstructorThebook'spresentationisinformal.Itemphasizesnotformulasbutunderstanding,interpretation,andappli-cability.Inmyopinion,formalismandproofareperfecttoolsandunavoidableforprofessionalmathemati-ciansbutareoflimitedvalueforstudents,eventosomeextentforundergraduatestudentsofmathematics.Someproofsoftheoremsaregiven,forothersinformalreasonings,butoftenstudentshavetorelyonevi-dencefromtheexamplesandsimulationsgiven.Itrytoavoidusingcomplicatedformulas.WhenIpresentaformula,Itrytoexplainitinwords.Ithelpsstudentstoconcentrateonconcreteexampleswithconcrete ii“Alles”—2014/5/8—11:19—pagexviii—#18iiiiiixviiiPrefacenumbers.Whenthebookintroducesparameters,theanalysiscarriesthemforwardasarststeptowardsabstraction.Althoughthebookiselementaryandavoidsformalism,itmaynotbeeasyreading.Somebooksconsideronly2by2bimatrixgamesorsimplereal-lifesituations,buttheexamplesinthisbookaresometimescomplex.Inmyopinion,thepowerofmathematicscannotbeappreciatedbylookingonlyatsmallexamples.Inthemlittlemathematicsisneededandcommonsenseshowsthebeststrategy.Thestudentswhousethisbookwillusetablesofnumberswithtentotwentyrowsandcolumns.Althoughthisrequiresmoreeffortthansmallerexamples,theExcelsheetswillhelpwiththecalculations,andintheendtheresultsshouldjustifytheextraeffort.Studentswillseethatmathematicaltheorysometimesproducesunexpectedresults.Thoughtheexamplesaresometimescomplex,theyarestillfarremovedfrommostrealgamesorfromseriousmodeling.Itrytomakethisclearthroughoutthebook.Thatdoesnotmeanthatnolessonscanbedrawnfromthem!Itiscommonknowledgethatmathematicscanbelearnedonlybydoing.Thebookcontainssixtyorsoprojectsthataskforthesolutionofagameandthepresentationofit.Experimentingandgoingbeyondwhatisdescribedinthetextisinmyopinioncrucialformasteringthematerial.Someprojectsaredifcultandsomeareopen-ended.Itellmystudentstodoasmuchastheycan,andtodothatwell!Figure1showshowthechaptersarerelated.Theeighttheorychaptersaresquares,theExcelisadiamond,thevehistorychaptersarestars,andtheexamplechaptersarecircles.ThebackboneofeverycoursewouldbethetheorychaptersandtheExcelchapter,chapters1,2,6,8,12,16,22,24,27,and32.Theyshouldinmyopinionbecoveredineverycourseongametheory.Someofthehistorychapters,chapters13,18,21,28,and39,couldbeassignedasreadingandmaterialforclassdiscussion.Figure1.StructureofthechaptersEachexamplechapterisattachedbyalinetotheearliesttheorychapterafterwhichitcanbecovered.Someoftheexamplechaptersformpairs,likeElectionandElectionII,whichareenclosedbydashedlines.IwouldnotrecommendcoveringapartIIchapterwithouthavingcoveredpartI.Therearetwolargergroups,thelocationgamesofchapters4,5,and11,andthepokerchapters25,31,36,37,38.InthesecasesIwouldrecommendatleastoneofchapters4or5beforechapter11,andcoveringchapter25beforecoveringanyofchapters31,36,37,or38.Asmoretoolsbecomeavailable,theexamplechaptersbecomemorecomplexanddifcult,solaterchap-tersaregenerallymoredifcultthanearlyones.InFigure1Iindicatethedifcultyoftheexamplechaptersbyshowingthemaboveorbelowthetheorybackbone.TheexamplechaptersbelowthebackboneIconsider ii“Alles”—2014/5/8—11:19—pagexix—#19iiiiiiPrefacexixtobeeasier,thoseabovemoredifcult.Butsometimesitispossibletoskipthedifcultparts.Forexample,chapters4,5,and11allcontainproofsthatrelategamestothestructureofgraphs,andthismaybeofinteresttomathematicsorcomputersciencemajors.Butthegamesprovideallreaderswithinstructiveexamplesofsimultaneousandsequentialgames.WhenIteachmycourse,Iusuallycoveratleastoneexamplechapteraftereachtheorychapter.Forageneraleducationmathematicscourseanappropriatesetmaybechapters3,4(justSection4.1),7,10,17,23,26,30,and35.Thoughthepokerchapters31,36,and37,arecomplex,Iusuallycoveroneoftheminmycourse,sinceIendthesemesterwitharobotpokertournamentsimilartotheonedescribedinchapter38.Thetournamentgetsthestudent'sattentionandisusuallyalotoffun!Instructorscanmodifytheappletstoruntheirowntournamentsandevenvarytherulesofthegame.AcknowledgementsEarlyversionsofthisbookwereusedasanonlinetextforsevenclassesatFranklinCollege(nowFranklinUniversity)between2007and2013.Iwanttothankallmystudentsforghtingtheirwaythroughtheseoftenincompleteandimperfectpages,andforthevaluablefeedbacktheygaveme.Iwanttothankinpar-ticularSamyAl-Ansari,NurbekAlmakuchukov,MeghanCanale,EricGors,RuthKappes,MatejKristic,GrigoreTocitu,KaramZaidkilani,ConnerCrawford,ChikaraDavis,JamesJasper,ArtyomMachekin,Nico-lausOwens,MeganAustin,DylanBontempo,MaijaButler,AndrianaFriel,ReneMusech,SannanPanaruna,AaronBrinckerhoff,CaitlinCurtis,MihaelDjambazov,MariahGrubb,SophieWinckel,NathanAyers,Jen-naleeBauman,AlexBromwell,StephanieErichsen,SachintGoel,SegevGoldberg,AndreaHak,MichaelRadomile,CooperStoulil,JenniferByram,ArnulfoHermes,ConnorMcCormack,BenjaminSmick,PaigeStumbough,EmilAkhmetov,(Kevin)YikfaiChan,ToddGilbert,IshanHill,AnnaHixson,AdrianLewis,ArianeMottale,LilyRybarczyk,AdamTaylor,andSenyaWebster.IalsowanttoexpressmygratitudetoFranklinUniversitySwitzerlandforgrantingmereleasedtimeduringtheyearsIhaveworkedonwritingthisbook.IamgratefultoJerryBryceandtheClassroomResourceMaterialsBoardofEditors,andwouldthankthefollowingmembersforreviewingmymanuscript:PhilMummert,PhilStrafn,SusanStaples,HollyZullo,SisterBarbaraReynolds,CynthiaWoodburn,andDianeHerrmann.Theirremarksgreatlyhelpedtoimprovethebook.ButmyverydeepestthanksgototheCRMEditorJerryBryceforgoingverycarefully,sentencebysentence,overthetextandcorrectingtypos,badEnglish,andpointingoutthemanycasesofunclearnessinearlierversionsofthemanuscript.Anyerrorsthatremainaremysoleresponsibility.WithoutJerry,thebookwouldnothavebeenpossible.ManythanksalsotoUnderwoodDudleyforcopyeditingthemanuscriptverycarefullyandgreatlyim-provingthemanuscript.IalsowanttothanktheMAAacquisitionseditorsDonAlbers,ZavenKarian,andSteveKennedy,andthewholeMAAeditingandproductionteam,inparticularCarolBaxterandBeverlyRuedi,forhelpwiththeproductionprocess. ii“Alles”—2014/5/8—11:19—pagexx—#20iiiiii ii“Alles”—2014/5/8—11:19—page1—#21iiiiiiCHAPTER1Theory1:Introduction1.1What'saGame?Everychildunderstandswhatgamesare.Whensomeoneoverreacts,wesometimessay“it'sjustagame.”Gamesareoftennotserious.Mathematicalgames,whicharethesubjectofthisbook,aredifferent.Itwasthepurposeofgametheoryfromitsbeginningsin1928tobeappliedtoserioussituationsineconomics,politics,business,andotherareas.Evenwarcanbeanalyzedbymathematicalgametheory.Letusdescribetheingredientsofamathematicalgame.RulesMathematicalgameshavestrictrules.Theyspecifywhatisallowedandwhatisn't.Thoughmanyreal-worldgamesallowfordiscoveringnewmovesorwaystoact,gamesthatcanbeanalyzedmathe-maticallyhavearigidsetofpossiblemoves,usuallyallknowninadvance.OutcomesandpayoffsChildren(andgrown-upstoo)playgamesforhoursforfun.Mathematicalgamesmayhavemanypossibleoutcomes,eachproducingpayoffsfortheplayers.Thepayoffsmaybemonetary,ortheymayexpresssatisfaction.Youwanttowinthegame.UncertaintyoftheOutcomeAmathematicalgameis“thrilling”inthatitsoutcomecannotbepredictedinadvance.Sinceitsrulesarexed,thisimpliesthatagamemusteithercontainsomerandomelementsorhavemorethanoneplayer.DecisionmakingAgamewithnodecisionsmightbeboring,atleastforthemind.Runninga100meterracedoesnotrequiremathematicalskills,onlyfastlegs.However,mostsportgamesalsoinvolvedecisions,andcanthereforeatleastpartlybeanalyzedbygametheory.NocheatingInreal-lifegamescheatingispossible.Cheatingmeansnotplayingbytherules.If,whenyourchessopponentisdistracted,youtakeyourqueenandputitonabettersquare,youarecheating,asinpoker,whenyouexchangean8inyourhandwithanaceinyoursleeve.Gametheorydoesn'tevenacknowledgetheexistenceofcheating.Wewilllearnhowtowinwithoutcheating.1.2Game,Play,Move:SomeDenitionsThecompletesetofrulesdescribesagame.Aplayisaninstanceofthegame.Incertainsituations,calledpositions,aplayerhasdomakeadecision,calledamoveoranaction.Thisisnotthesameasstrategy.Astrategyisaplanthattellstheplayerwhatmovetochooseineverypossibleposition.Rationalbehaviorisusuallyassumedforallplayers.Thatis,playershavepreferences,beliefsabouttheworld(includingtheotherplayers),andtrytooptimizetheirindividualpayoffs.Moreover,playersareawarethatotherplayersaretryingtooptimizetheirpayoffs.1 ii“Alles”—2014/5/8—11:19—page2—#22iiiiii21.Theory1:Introduction1.3ClassicationofGamesGamescanbecategorizedaccordingtoseveralcriteria:Howmanyplayersarethereinthegame?Usuallythereshouldbemorethanoneplayer.However,youcanplayroulettealone—thecasinodoesn'tcountasplayersinceitdoesn'tmakeanydecisions.Itcollectsorgivesoutmoney.Mostbooksongametheorydonottreatone-playergames,butIwillallowthemprovidedtheycontainelementsofrandomness.Isplaysimultaneousorsequential?Inasimultaneousgame,eachplayerhasonlyonemove,andallmovesaremadesimultaneously.Inasequentialgame,notwoplayersmoveatthesametime,andplayersmayhavetomoveseveraltimes.Therearegamesthatareneithersimultaneousnorsequential.Doesthegamehaverandommoves?Gamesmaycontainrandomeventsthatinuenceitsoutcome.Theyarecalledrandommoves.Doplayershaveperfectinformation?Asequentialgamehasperfectinformationifeveryplayer,whenabouttomove,knowsallpreviousmoves.Doplayershavecompleteinformation?Thismeansthatallplayersknowthestructureofthegame—theorderinwhichtheplayersmove,allpossiblemovesineachposition,andthepayoffsforalloutcomes.Real-worldgamesusuallydonothavecompleteinformation.Inourgamesweassumecompleteinfor-mationinmostcases,sincegamesofincompleteinformationaremoredifculttoanalyze.Isthegamezero-sum?Zero-sumgameshavethepropertythatthesumofthepayoffstotheplayersequalszero.Aplayercanhaveapositivepayoffonlyifanotherhasanegativepayoff.Pokerandchessareexamplesofzero-sumgames.Real-worldgamesarerarelyzero-sum.Iscommunicationpermitted?Sometimescommunicationbetweentheplayersisallowedbeforethegamestartsandbetweenthemovesandsometimesitisnot.Isthegamecooperativeornon-cooperative?Evenifplayersnegotiate,thequestioniswhethertheresultsofthenegotiationscanbeenforced.Ifnot,aplayercanalwaysmovedifferentlyfromwhatwaspromisedinthenegotiation.Thenthecommunicationiscalled“cheaptalk”.Acooperativegameisonewheretheresultsofthenegotiationscanbeputintoacontractandbeenforced.Theremustalsobeawayofdistributingthepayoffamongthemembersofthecoalition.ItreatcooperativegamesinChapter35.StudentActivityPlaytenroundsintheappletsforeachoneofthegamesLisaGame,QuatroUno,andAuction.Inthersttwoyoucanplayagainsta(well-playing)computer,butinthethirdthecomputerservesonlyasauctioneerandyouneedtondanotherhumanplayer.Categorizeeachgameassimultaneousorse-quentialorneither,anddeterminewhetherrandomnessisinvolved,whetherthegamehasperfectinformation,andwhetheritiszero-sum.Determinethenumberofplayersineachgame,andjustifyyourconclusions.ModelingNoteAnalyzinggameslikeparlorgamesorcasinogamesmayseemtobeenoughmotivationtodevelopatheoryofgames.However,gametheoryhashigheraims.Itprovidestoolsthatcanbeappliedinmanysituationswheretwoormorepersonsmakedecisionsinuencingeachother.Amodelisanabstract,oftenmathematical,versionofreality.Inthisbookamodelisagame,whichissupposedtoyieldsomeinsightintoareal-worldsituation.Itisimportantnottoconfusethemodelwithreality—inrealitytherearealmostnevertotallystrictrulesandplayersalmostalwayshavemoreoptionsthantheythinktheyhave,morethanwhatthemodelallows.Inthisbookwealsowilltrytomodelsomereal-worldsituationsasgames,buttheapproachtakeniscautious.Wheneverwetrytomodelareal-lifesituation,wewill ii“Alles”—2014/5/8—11:19—page3—#23iiiiii1.3.ClassicationofGames3discussindetailtheassumptionsofthemodelandwhethertheconclusionsfromthemodelarerelevant.Whethergametheorycanbeusefulinreallifeissomethingforeachreadertodecide.Exercises1.InEnglishauction,anitemisauctioned.Peopleincreasebidsinincrementsof$10,andtheplayergivingthehighestbidgetstheitemforthatamountofmoney.Givereasonswhytheauctioneerwouldbeconsideredaplayerofthegame,orreasonswhyheorshewouldnot.Doesthegamecontainrandommoves?Isitzero-sum?Wouldareal-worldartauctionhavecompleteinformation?2.Inroulette,wouldthecroupierbeconsideredtobeaplayer?Doesthegamecontainrandommoves?Isitzero-sum?Canplayersincreasetheirchancesofwinningiftheyformacoalitionanddiscusshowtoplaybeforeeachround?3.Inthewell-knowngamerock,scissors,papergame,howmanyplayersarethere?Isitsimultaneous,orsequential,orneither,and,ifitissequential,doesithaveperfectinformation?4.Forpoker,discussnumberofplayers,whetheritissequentialorsimultaneous,orneither,andifitisse-quential,whetherithasperfectinformation.Discusswhethertherearerandommoves.Iscommunicationallowedinpoker?5.Forblackjackdiscussitsnumberofplayers;whetheritissequentialorsimultaneous,orneither;andifitissequential,whetherithasperfectinformation.Discusswhethertherearerandommoves.Iscommunicationallowedinblackjack?6.It'slateafternoonandyouareinatraintravelingalongacoastline.Fromtimetotimethetrainstopsinvillages,someofthemnice,someofthemugly,andyoucanevaluatethenicenessofthevillageimmediately.Thebenetofaneveningandnightspentatthatvillagedependsonlyonitsniceness.Youwanttogetoffatthenicestvillage.Unfortunatelyyoudon'tknowhowmanyvillagesarestilltocome,andyouknownothingabouthowvillagesinthiscountrynormallylook.Worse,youarenotabletoaskanybody,sinceyoudon'tspeakthelanguageofthecountry.Youalsoknowthatsome(unknown)timeintheeveningthetrainwillreachitsdestinationwhereyouwillhavetostaywhetheritisniceornot.Explainthefeaturesofthisgame,withemphasisontheinformationalissues.Howwouldyouplayit?Giveareasonforyourstrategy.Commentonwhetherwehavecompleteorincompleteinformationhere,andwhy.(InitiallyIformulatedthisexampleintermsofmarriageinasocietywheredivorceisimpossible,butIsawthatthisisadifferentgame.Couldyougivesomeargumentswhy?)7.Inthismorerealisticversionofthegamesupposethatyouknowthatthetrainwillstopintenvillagesbeforeitreachesitsdestination.Howwouldyouplaynow?Commentonwhetherwehavecompleteorincompleteinformationhere,andjustifyyourcomment. ii“Alles”—2014/5/8—11:19—page4—#24iiiiiiCHAPTER2Theory2:SimultaneousGamesInhisstory“JewishPoker”thewriterEphraimKishondescribeshowamancalledErvinkeconvincesthenarratortoplayagamecalledJewishPokerwithhim.“Youthinkofanumber,Ialsothinkofanumber”,Ervinkeexplains.“Whoeverthinksofahighernumberwins.Thissoundseasy,butithasahundredpitfalls.”Thentheyplay.IttakesthenarratorsometimeuntilherealizesthatitisbettertoletErvinketellhisnumberrst.[K1961]Obviouslythisisagamethatisnotfairunlessbothplayersplaysimultaneously.Inthischapterwewillstartourjourneythroughgametheorybyconsideringgameswhereeachplayermovesonlyonce,andmovesaremadesimultaneously.Thegamescanbedescribedinatable(calledthegame'snormalform).Thenwediscussapproachesthatallowtheplayerstodecidewhichmovetheywillchoose,culminatingwiththefamousNashequilibrium.2.1NormalForm—BimatrixDescriptionImagineyouwanttodescribeasimultaneousgame.Weknowthateachplayerhasonlyonemove,andthatallmovesaremadesimultaneously.Whatelsedoweneedtosay?First,wemuststipulatethenumberofplayersinthegame.Second,wemustlistforeachplayerallpossiblemoves.Differentplayersmayhavedifferentrolesandmayhavedifferentoptionsformoves.Weassumethateachplayerhasonlynitelymanyoptions.Playerssimultaneouslymaketheirmoves,determinetheoutcomeofthegame,andreceivetheirpayoffs.Weneedtodescribethepayoffsforeachoutcome.Howmanyoutcomesarepossible?Eachcombinationofmovesoftheplayersgeneratesadifferentout-come.Iftherearenplayers,andplayer1hask1possiblemoves,player2hask2possiblemoves,andsoon,thentherearek1k2


knpossibleoutcomes.Foreach,nnumberswoulddescribethepayoffsforplayer1,player2,andsoon.Ingameswhereeachplayerhasinnitelymanyoptions,wemayusemethodsofcalculusforfunctionswithtwovariables,butsuchgamesarenotdiscussedinthisbook.WedescribesimultaneousgameswithrandomnessinChapter12.2.1.1TwoPlayersHereisanexampleofasimultaneous2-playergame:Example1ADVERTISING:Twocompaniesshareamarket,inwhichtheycurrentlymake$5,000,000each.Bothneedtodeterminewhethertheyshouldadvertise.Foreachcompanyadver-tisingcosts$2,000,000andcaptures$3,000,000fromthecompetitorprovidedthecompetitordoesn'tadvertise.Whatshouldthecompaniesdo?4 ii“Alles”—2014/5/8—11:19—page5—#25iiiiii2.1.NormalForm—BimatrixDescription5Let'scallthetwocompaniesAandB.Ifbothdon'tadvertise,theyget$5,000,000each.Ifbothadvertise,bothlowertheirgainto$3,000,000.IfAadvertises,butBdoesn't,Agets$6,000,000andBonly$2,000,000,andconverselyifBadvertisesandAdoesn't.Thepayoffpatternisshowninthefollowingtable.Thenumbersareinmillionsofdollars.TherowscorrespondtotheoptionsofplayerA,andthecolumnscorrespondtotheoptionsofplayerB.TheentriesarepayoffforAandpayoffforBprovidedthecorrespondingoptionsarechosen,separatedbyacomma.BadvertisesBdoesn'tadvertiseAadvertises3,36,2Adoesn'tadvertise2,65,5Wheneverwehavetwoplayers,weoftennamethemAnnandBeth.AssumeAnnhask1optionsandBethhask2options.WewanttodisplaythedifferentpayoffsforAnnandBeth,dependingonthedifferentchoicestheyhave.Eachofthek1
k2outcomeshaspayoffsforAnnandBethattached.Usuallythisisvisualizedinatable,thenormalformwithnrows,correspondingtoAnn'soptions,andmcolumns,correspondingtoBeth'soptions.Suchatableiscalledanmbimatrix.TheentriesinthecellsarepayoffsforAnnandBeth,separatedbyacomma.2.1.2TwoPlayers,Zero-sumAgameiscalledzero-sumifthesumofpayoffsequalszeroforanyoutcome.Thatmeansthatthewinningsofthewinningplayersarepaidbythelossesofthelosingplayers.Forzero-sumtwo-playergames,thebimatrixrepresentationofthegamecanbesimplied:thepayoffofthesecondplayerdoesn'thavetobedisplayed,sinceitisthenegativeofthepayoffoftherstplayer.Example2AssumeweareplayingROCK-SCISSORS-PAPERforonedollar.ThenthepayoffmatrixisRockScissorsPaperRock011Scissors101Paper110Therstcellsays“0”,whichstandsfor“0;0”apayoffof0forbothplayers.Thesecondcellentryof“1”shouldbereadas“1;1”,apayoffof1forAnnwhichhastobepaidbyBeth,thereforeapayoffof1forBeth.2.1.3ThreeorMorePlayersIfwehavemorethantwoplayers,weneedanothersystematicwaytogeneratetheneededk1
k2



kncellscorrespondingtothedifferentoutcomes,intowhichwewritethenpayoffsforthenplayers.Hereisanexample:Example3LEGISLATORS'VOTE:Threelegislatorsvotewhethertheyallowthemselvesaraiseinsalaryof$2000peryear.Sincevotersareobservingthevote,thereissomelossoffaceforalegislatortovoteforaraise.Let'sassumethatthelegislatorsestimatethatlossoffaceisworth$1000peryear.Whathappensifallthreevoteatthesametime?(Thisgameisavariantofthegamedescribedin[K2007]). ii“Alles”—2014/5/8—11:19—page6—#26iiiiii62.Theory2:SimultaneousGamesThisisasimultaneousthree-playergame.Itisbestvisualizedwithtwomatrices.PlayerAchoosesthematrix,Bchoosestherow,andCchoosesthecolumn.Thepayoffs(inthousandsofdollars)areAvotesforaraiseAvotesagainstaraiseCvotesCvotesforraiseagainstitBvotes1,1,11,1,2forraiseBvotes1,2,1-1,0,0againstCvotesCvotesforraiseagainstitBvotes2,1,10,-1,0forraiseBvotes0,0,-10,0,0against2.1.4SymmetricGamesAllourexamplessofararesymmetric:Allplayershavethesameoptions,andifthetwoplayersinterchangetheirmoves,thepayoffsarealsointerchanged.Moreformally,fora2-playergame,letm1;m2bemovesandleta.m1;m2/andb.m1;m2/beAnn'sandBeth'spayoffsifAnnplaysm1andBethplaysm2.Thena.m1;m2/Db.m2;m1/andb.m1;m2/Da.m2;m1/forsymmetricgames.Thatmeansthattheentriesinthejsrowandtheiscolumnisobtainedfromtheentriesintheisrowandjscolumnbyinterchang-ingthepayoffs.Forsymmetric3-playergames,a.m1;m2;m3/Db.m2;m1;m3/Db.m3;m1;m2/Dc.m2;m3;m1/Dc.m3;m2;m1/,andsoon.Symmetricgamesarefairbydesign,givingthesamechancestoeveryplayer.2.2WhichOptiontoChooseItisusefultodescribesimultaneousgamesbyabimatrix,butwhatplayerswantisadviceonhowtoplay.Gametheoryshould(andwillinsomecases)provideplayerswithamechanismtondwhichmoveisbest.Themechanismswouldrefertothebimatrixonly,thesolutionwouldbethesamenomatterwhetherwefaceacasinogameorawar,providedthecorrespondingmatricesarethesame.Theessenceofthegameliesinthenumbersinthebimatrix.Suchmechanismsarethecontentofthissection.Wewillgivethreeorfourofthem.Likeadvicefromwell-meaninguncles,theyallhaveagoodandconvincingpoint,butsincetheyconcentrateondifferentfeaturesofthegame,theydon'talwaysleadtothesameconclusion.Wewilldiscussthemrstseparatelybeforeinvestigatingtherelationsbetweenthem.2.2.1MaximinMoveandSecurityLevelSomepeoplealwaysexpecttheworst.Nomatterwhatsheplays,aplayer(let'scallherAnn)mayassumethattheotherplayerswillalwaysrespondwithmovesthatminimizeAnn'spayoff.Thismaybejustiedinatwo-playerzero-sumgameifAnnissopredictablethattheotherplayeralwaysanticipatehermove.Inothercasesthebeliefbordersonparanoia,sincetheotherplayerswillnotbeinterestedinharmingAnnbutinsteadwanttomaximizetheirpayoffs.Still,pessimisticAnnwillevaluateherstrategiesinlightoftheworstexpectedcase.Shewouldconcentrate,foranyofheroptions,onhersmallestpossiblepayoff.Ifshebelievesthatthisiswhatshewouldget,thenAnnwouldchoosetheoptionwithhighestvalue.Thisvalueiscalledthemaximinvalueorsecuritylevel.TheoptionAnnwillplayiscalledamaximinmove(strategy),sinceitmaximizestheminimumpossiblepayoff.Playingthemaximinmove,theplayercanguaranteeapayoffofatleastthemaximinvalue,nomatterhowtheothersareplaying.Tochoosethemaximinmove,theplayerdoesn'thavetoknowthepayoffsoftheotherplayers.IntheADVERTISINGexample,companyAmayfearthatcompanyBwilladvertisetooifAadvertises,yieldingapayoffof3forA.IfcompanyAdoesnotadvertise,theworstthatcouldhappenwouldbecompany ii“Alles”—2014/5/8—11:19—page7—#27iiiiii2.2.WhichOptiontoChoose7Badvertisingwithpayoffof2forA.ThereforecompanyAwouldadvertisetomaximizetheworstpossiblepayoff.IntheLEGISLATORS'VOTEexample,theworstthatcouldhappenifAvotesforaraiseisthatbothothersvoteagainst,leavingAwithapayoffof1000.IfAvotesagainstaraise,intheworstcase(actuallyinthreeoffourcases)Agetsapayoffof0,whichismorethanintheothercase.ThereforeAwouldvoteagainstaraiseifusingthemaximinprinciple.Inatwo-playergametherstplayer,Ann,wouldlookattherowsofthebimatrixandineachrowhighlightthecellwithherlowestpayoff.Thenshewouldselecttherowwiththehighestnumberhighlighted.Inthesameway,thesecondplayer,Beth,whenplayingthemaximinstrategywouldmarkineachcolumnthecellwithlowestpayoffforBeth,andthenselectthecolumnwiththehighestnumbermarked.Howdowetreatties,iftwoormorerowshavethesameminimumpayoffforAnn?Anncouldchooseoneofthesemoves,oralternaterandomlybetweensuchmoves.ThelatterleadstomixedstrategiesthatarecoveredinChapter27.2.2.2DominatedMovesAmove,M1,forAnnstrictlydominatesanotherM2,ifM1alwaysresultsinahigherpayoffforAnnthanM2.Arationalplayerwouldneverplayamovethatisstrictlydominatedbyanotherone.Dominationdoesn'ttellwhattoplaybutratherwhatnottoplay.IntherarecasewhereoneofAnn'smovesstrictlydominatesallherothermoves,thiswouldturnintopositiveadvicetoplaythemovedominatingallothermoves.IntheADVERTISINGexample“advertising”strictlydominates“notadvertising”forbothcompanies.Thereforebothcompanieswilladvertise,whenapplyingthismechanism.Itisnocoincidencethattheadvicegivenbythemaximinmechanismandtheadvicegivenbytherulenottoplaystrictlydominatedmovesarethesameforthisexample.Actuallyaplayer'smaximinmoveisneverstrictlydominatedbyanyofherothermoves.Adviceforplayerscouldgofurtherthantodisregardstrictlydominatedmoves.Inparticular,ifplayerAnnbelievesthatotherplayerswouldalsoobeythisrule,thenwemaydisregardallstrictlydominatedmovesinthegame,notonlyforAnnbutforallotherplayers.However,thisassumptionabouttheotherplayers'behaviorisnotautomatic.Itassumesthatallplayersarerationalandcleverorexperiencedenough.Undertheassumptionthatallplayersacceptthisbeliefintherationalityandsophisticationofallplayers,weknowthatallplayersreducethegamebyeliminatingallstrictlydominatedmoves.Then,inthereducedgame,strictdominationmayoccurwhereithadnotbefore,andthesameroundofeliminationscouldbedonetoreducethegamefurther.Theprocessofrepeatedlyreducingthegame,aswellasitsresult,agamethatcannotbereducedanyfurthersincetherearenostrictlydominatedmoves,isdenotedbyIESD—iteratedeliminationofstrictlydominatedmoves.ExceptincaseswheretheIESDresultisagamewithjustoneoptionforAnn,IESDisamethodofexcludingmovesratherthantellingwhatmovetochoose.HereisasuccessfulapplicationoftheIESDprocedure:Example4TWOBARS:Eachoneoftwobarschargesitsownpriceforabeer,either$2,$4,or$5.Thecostofobtainingandservingthebeercanbeneglected.Itisexpectedthat6000beerspermontharedrunkinabarbytourists,whochooseoneofthetwobarsrandomly,and4000beerspermontharedrunkbynativeswhogotothebarwiththelowestprice,andsplitevenlyincasebothbarsofferthesameprice.Whatpriceswouldthebarsselect?[S2009]Thegame,asallgamesconsideredsofar,issymmetric.Letmeillustrateinoneinstancehowtocomputethepayoffs.IfbarAcharges$2andbarBcharges$4,thenallnativeswillchoosebarA. ii“Alles”—2014/5/8—11:19—page8—#28iiiiii82.Theory2:SimultaneousGamesThereforebarAwillserve4000beerstothenatives,and3000beerstotourists,serving7000beersintotal,making7000
2D14000dollars.BarBwillonlyserve3000beerstotourists,making3000
4D12000dollars.Thepayoffmatrix,withvaluesinthousandsofdollars,is245210,1014,1214,15412,1420,2028,15515,1415,2825,25Foreachbar,move“4”strictlydominatesmove“2”,thereforewecouldeliminatebothmoves“2”togetthereducedgame:45420,2028,15515,2825,25Now,butnotbeforetheelimination,move“4”strictlydominatesmove“5”.Thereforeweeliminatethesemovesforbothplayersaswellandarriveatagamewithonlyoneoption,“4”,foreachplayer,andapayoffof$20000foreach.Thereforebothplayerswillchoose$4asthepriceofthebeer.Aweakerconditionisweakdomination.Ann'smoveweaklydominatesanotheroneofhermovesifityieldsatleastthesamepayoffforAnninallcasesgeneratedbycombinationsofmovesoftheotherplayers,andinatleastonecaseanevenbetterpayoff.Sotheweaklydominatingmoveisneverworsethantheweaklydominatedone,andsometimesitisbetter.Thecommonwisdomisthatiteratedeliminationofweaklydominatedmoves,IEWDisnotsomethingthatshouldperformedautomatically.Weaklydominatedmovesmaystillbeplayed,inparticularincaseswheretheweaknessoftheweaklydominatedmoveappearsinacombinationwithotherplayer'smovesthatareknownnottobeplayedbythem.ThisopinionisalsobasedondifferentbehaviorofNashequilibria(discussedinSection2.4)underIEWDandIESD.2.2.3BestResponseAssumeyouwillplayaone-roundsimultaneousgameagainstyourfriendtomorrow.Yourfriendhasbeenthinkingabouthermove,arrivesonadecisionwhatmovetoplay,andwritesitonapieceofpapersoasnottoforgetit.Yougetalookatthispaperwithoutyourfriendnoticingit.Thegamethuschangesfromsimultaneoustosequentialwithperfectinformation.Themoveyouplayundertheseconditionsiscalledthebestresponsetothemoveofyourfriend.Letusstartwithtwoplayers.Ann'sbestresponsetoBeth'smoveMisthemovethatyieldsthehighestpayoffforAnn,givenBeth'smoveM.Theremaybeseveralbestresponsestoagivenmove.TondAnn'sbestresponsetoBeth'smoveM,wedon'tevenhavetoknowBeth'spayoffs.Youndthebestresponsesfortherstplayer's(Ann's)movesbylookingattherowsofthebimatrixonebyoneandselectingineachrowthecellwherethesecondentryisamaximum.Thelabelofthecorrespondingcolumnisthebestresponsetothemovecorrespondingtothatrow.Inthesameway,tondbestresponsesagainstthesecondplayer's(Beth's)movesweconsiderthecolumnsandpickineachcolumnthecellwithmaximumrstentry.Thelabelofthecorrespondingrowisthecorrespondingbestresponseforthemovecorrespondingtothatcolumn.IntheADVERTISINGexample,thebestresponsetoadvertisingistoadvertise,andthebestresponsetonotadvertisingisalsotoadvertise.Thisholdsforbothplayers,sincethegameissymmetric.IntheTWOBARSexample,thebestresponsetoapriceof“2”isapriceof“5”,thebestresponsetoapriceof“4”isapriceof“4”,andthebestresponsetoapriceof“5”isapriceof“4”.Thegameissymmetric. ii“Alles”—2014/5/8—11:19—page9—#29iiiiii2.2.WhichOptiontoChoose9Example5Letusgiveanasymmetricexample.AssumeAnnhasfourmoves,A1,A2,A3,A4,andBethhasthreeB1,B2,andB3.Thepayoffbimatrixofthisnonzero-sumtwo-persongameisB1B2B3A11,32,21,2A22,32,32,1A31,11,23,2A41,23,12,3WendBeth'sbestresponsetoAnn'smoveA1byndingthelargestsecondvalue(Beth'spayoff)intherstrow,whichisunderlined.ThatimpliesthatBeth'sbestresponsetoAnn'smoveA1ismoveB1.Inthesamewayweunderlinethehighestsecondvaluesinotherrows,andconcludethatBeth'sbestresponsestoAnn'smoveA2arebothmovesB1andB2,Beth'sbestresponsestomoveA3arebothmovesB2andB3,andthatBeth'sbestresponsetomoveA4ismoveB3.TondAnn'sbestresponses,weunderlineineachcolumnthehighestrst(Ann'spayoff)entry.ThereforeAnn'sbestresponsetoBeth'smoveB1isA2,Ann'sbestresponsetoB2isA4,andAnn'sbestresponsetoB3isA3.BestResponseforThreePlayersBestresponsesmakealsosenseforgameswiththreeormoreplayers.FordetectingthebestresponsemovesofBeth,welookatthesecondentries(Beth'spayoffs)ineachcolumnandmarkthehighestvalue.Todetectthebestresponsemovesforthethirdplayer(let'scallherCindy)welookatthethirdentriesoftherowsandmarkineachrowthehighestentry.ForAnnthemethodisalittlemorecomplicatedtoexplain.Herewelookatrstentriesonly,andcomparecellshavingthesamepositioninthedifferentmatrices,as“upperleft”,forinstance.Example6Let'sndbestresponsesinanexampleofasimultaneousthree-persongamewhereeachplayerhastwooptions,AnnhasthemovesA1andA2,BethhasB1andB2,andCindyhasC1andC2.AssumethepayoffsareA1A2C1C2B10,2.1,01,1.1,0.1B21,0,10,1,1.1C1C2B10.1,1.1,11.1,0.1,0:9B20:9,1,00.1,2,0.1Becausethehighestsecondentryintherstcolumnis2.1,itisunderlined.Thehighestsecondentryinthesecondcolumnis1.1,inthethirdcolumn(rstcolumnofthesecondmatrix)1.1,andinthefourthcolumn2,sotheyareunderlined.ForCindy'sbestresponses,thehighestthirdentryintherstrowoftherstmatrixis0.1.Thehighestthirdentryinthesecondrowoftherstmatrixis1.1.Forthesecondmatrix,thehighestthirdentryintherstrowis1,andinthesecondrow0.1.ForAnn,thehighestrstentryofupper-leftcellsinthetwomatricesis0.1,thehighestrstentryofupper-rightcellsis1.1,andweget1respectively0.1forthelower-leftrespectivelylower-rightcells.2.2.4NashEquilibriaInthissectionwewillidentifyoutcomes—combinationsofmovesforeachplayer—thataremorelikelytooccurthanothers.AnoutcomeiscalledapureNashequilibriumprovidednobodycangainahigherpayoffbydeviatingfromthemove,whenallotherplayerssticktotheirchoices.Ahigherpayoffforaplayermaybepossible,butonlyiftwoormoreplayerschangetheirmoves.Anoutcome,acombinationofmoves,isapure ii“Alles”—2014/5/8—11:19—page10—#30iiiiii102.Theory2:SimultaneousGamesNashequilibriumifeachmoveinvolvedisthebestresponsetotheothermoves.AcellinthenormalformisapureNashequilibriumifeachentryismarked(underlinedinourexamples)asbeingthebestresponsetotheothermoves.NashequilibriawereintroducedbyJohnNasharound1950.Nashequilibriaareself-enforcingagreements.Ifsome(non-binding)negotiationhastakenplacebeforethegameisplayed,eachplayerdoesbest(assumingthattheotherplayerssticktotheagreement)toplaythenegotiatedmove.Inthersthalfofthebook,Nashequilibriawillbepure.Chapter27willintroducemixedNashequilibria.InExample5,therearetwoNashequilibria:.A2;B1/and.A3;B3/.InthesymmetricTWOBARSex-ample(4,4)istheuniquepureNashequilibrium.AsanotherexampleweconsiderthefamousPRISONER'SDILEMMA.PRISONER'SDILEMMAAdamandBobhaverobbedabankandbeenarrested.Theyareinter-rogatedseparately.AdamandBobhavetheoptiontoconfess(moveC)ortoremainsilent(moveS).Thepolicehavelittleevidence,andifbothremainsilenttheywillbesentencedtooneyearonaminorcharge.Thereforethepoliceinterrogatorsproposeadeal:ifoneconfesseswhiletheotherremainssilent,theoneconfessinggoesfreewhiletheotherissentencedtothreeyears.However,ifbothtalk,bothwillstillbesentencedtotwoyears.Ifeachplayer'spayoffis3minusthenumberofyearsservedinjail,wegetthefollowingpayoffbimatrix:SCS2,20,3C3,01,1Itseemsobviousthatbothshouldremainsilent,butthat'snotlikelytohappen.Eachplayer'smoveCstrictlydominatesmoveS.Furthermore,thebestresponsetomoveSisC,andthebestresponsetomoveCisalsomoveC,thereforethepair(C,C)—bothconfessingformstheuniqueNashequilibriumofthisgame.ThechoiceC—confessing—withpayoffsofonly1mayseemcounterintuitiveifnegotiationscantakeplaceinadvance,buttheirtermsarenon-bindingandcannotbeenforced.ItwouldbeuselesstoagreeonmoveSinadvance,sinceeachoftheplayerswouldfeelastrongurgetodeviate(cheat).Onlyifbindingagreementsarepossible,wouldbothagreeontheS-Scombination,reachingahigherpayoff.ThusPRIS-ONER'SDILEMMAgivesaparadoxicalresult.Playerswillplaymovesthatresultinlowerpayoffsforboththanarepossible.Thisisinpartbecausetherulesofthegamedonotallowbindingagreements.Noteverysimultaneousgamehasa(pure)Nashequilibrium.AnexampleisExample2ROCK-SCISSORS-PAPER.NextweconsideragamewithmorethanoneNashequilibrium:Example7BATTLEOFTHESEXES:Acouple,AdamandBeth,decideindependentlywhethertogotoasoccergameortotheballetintheevening.Eachpersonlikestodosomethingtogetherwiththeother,butthemanpreferssoccer,andthewomanprefersballet.Tosimplifythegame,weassumethatthetotalpayoffforeachplayeristhesumofthepayoffs(intermsofsatisfaction)ofbeingatthepreferredplace,whichgivesasatisfactionofcsatisfactionunits,and ii“Alles”—2014/5/8—11:19—page11—#31iiiiii2.2.WhichOptiontoChoose11beingtogetherwiththepartner,givingdsatisfactionunits.Wehavetwovariants,dependingonwhethercordislarger,theloworhighlovevariants.Thepayoffhereissatisfactioninsteadofmoney.Theassumptionoftheadditivityofsatisfactionissevere—satisfactioncouldjustaswellbemultiplicative,orsomemorecomplicatedfunctionofcandd.Itcouldevenbethatsatisfactioninoneareacouldinterferewithsatisfactionintheother.Thesatisfactionsmaydifferforbothpersons,oneappreciatingthepresenceoftheothermorethantheother,oronehavingaclearpreferenceforsoccerorballet,whentheotherisindifferent.Exampleswillbegivenintheexercises.Asthisexamplewasdevisedbeforetherewerecellphones,weassumethatnopreviouscommunica-tionispossible.Herearethepayoffbimatricesforbothvariants,whereAdamchoosestherowsandBethchoosesthecolumns.HighLoveversion,cD1,dD2LowLoveversion,cD2,dD1soccerballetsoccer3,21,1ballet0,02,3soccerballetsoccer3,12,2ballet0,01,3ThehighloveversionofBATTLEOFTHESEXEShastwoNashequilibria:(soccer,soccer)and(ballet,ballet).ForAdamchoosing“soccer”,Beth'sbestresponseis“soccer”.ForAdamchoosing“ballet”,Beth'sbestresponseis“ballet”.Also,Adam'sbestresponseforBethchoosing“soccer”is“soccer”,andhisbestresponseforBethchoosing“ballet”is“ballet”.ThelowloveversionhasoneNashequilibrium,namely(soccer,ballet):bothplayersgowheretheywanttogoanyway.ModelingNoteWemadeasimpleassumptionintheexample,namelythatthetotalpayoffforaplayeristhesumoftheutilitiesofcertainingredients.Inmanysituationswewillusethisapproach,sinceitissimpleandisthewaymoneyisadded.However,therearesituationswhereadditivityisnotappropriate.Oneassetmayestablishworthonlywhencombinedwithanotherasset,asaleftshoeandthecorrespondingrightshoe,ormoneyandfreetimeavailable.Inmanysituationseachhasrealvalueonlyincombi-nationwiththeother.GameswithmorethanonepureNashequilibriumaresometimescalled“coordinationgames”,sinceifpre-gamenegotiationsareallowed,theplayershavetoagreeononeofthem.ThehighloveversionofBATTLEOFTHESEXESisanexample.Inthiscase,theobviousquestionis:whichNashequilibriumisthebest?OneideaistoconcentrateonPareto-optimalNashequilibriaonly.ANashequilibriumisPareto-dominatedbyanotherNashequilibriumifeveryplayer'spayoffintherstoneissmallerorthesameasinthesecondone.NobodywouldobjecttomovetothesecondNashequilibrium.ANashequilibriumisPareto-optimalifitisnotPareto-dominatedbyanyotherNashequilibrium,exceptmaybebysomehavingexactlythesamepayoffs.IntheBATTLEOFTHESEXESexample,bothNashequilibriaarePareto-optimal.Forgameswithmorethantwoplayers,weusethemarking(underlining)procedureasdescribedinthesectiononbestresponses.ThenthecellswithallentriesunderlinedarethepureNashequilibria.InExample6,a3-playergame,wehavetwopureNashequilibria—thecellswhereallentriesareunder-lined,whereeachmoveisthebestresponsetothepairofmovesoftheothertwoplayers.Thesearethetriples(A2,B1,C1)and(A2,B2,C2).SoplayerAwillprobablychooseA2. ii“Alles”—2014/5/8—11:19—page12—#32iiiiii122.Theory2:SimultaneousGamesInourotherexampleofa3-playergame,LEGISLATORS'VOTE,letusunderlinethebestresponses:AvotesforaraiseAvotesagainstaraiseCvotesCvotesforraiseagainstitBvotes1,1,11,1,2forraiseBvotes1,2,11,0,0againstCvotesCvotesforraiseagainstitBvotes2,1,10,1,0forraiseBvotes0,0,10,0,0againstHerewehavefourpureNashequilibria:thethreeoutcomeswheretwolegislatorsvoteforaraiseandonevotesagainst,andtheonewhereallthreevoteagainst.ThefourthequilibriumisPareto-dominatedbytheotherthree,soitisnotPareto-optimalandisthereforelessimportantthantheotherthree.Thenextexample,a5-playergame,illustrateshowyoucandeterminewhetheranoutcomeisaNashequilibriumwhenyoudon'thaveabimatrixrepresentation:Example85KNIGHTS:Fiveknights,A,B,C,D,E,areelectingtheirleader.Eachonehasalistofpreferences.Examplesofpreferences,givenfromhighesttolowest,areA:A,D,E,C,BB:B,C,E,A,DC:C,E,D,B,AD:D,B,C,E,AE:E,C,B,A,D.Theyelectinrounds.Ineachround,eachknightsubmitsonename.Aknightiselectedifhegetsmorevotesthanalltheothers.Soeventwovotesmaysufceifnootherknightgetstwovotes.Ifnooneiselected,weproceedtothenextround.Therearetwoversions:EarlyCaseIftheknight'srstchoiceiselected,thisisapayoffof2forthatknight.Ifhissecondchoiceiselected,hispayoffis1.Ifnobodyiselectedandweproceedtothenextround,thepayoffis0.Ifhisthird,fourth,orfthchoiceiselected,hispayoffis1,2,or3.ExhaustedCaseTheknight'srst,second,andthirdchoicegivespayoffsof2,1,and0.Ifnooneiselectedandweproceedtothenextround,thepayoffis1.Ifhisfourthorfthchoiceiselected,hispayoffis2or3.Eachpreferencepatterndenesanewgame.Becauseeveryplayerhasveoptions,thereare5
5
5
5
5D3125outcomes.Wecouldrepresentthemwithpayoffsona5-dimensionalcube.Let'sinsteadlookatanoutcomeanddeterminewhetheritisaNashequilibriuminthetwoversionsofthegame.AssumeAvotesforA,BvotesforB,CvotesforC,DvotesforC,andEvotesforC.ThenCiselected,andthepayoffsforA,B,C,D,Eare2,1,2,1,1intheearlycasegame.KnightAisnothappybutstillhasnoreasontovotedifferently—ifhevotedforAorDinstead,Cwouldstillbeelected.ButthisoutcomeisnotaNashequilibrium,sinceD,knowingthevotingpatternoftheothers,wouldratherhavevotedforBtoobtainatieandapayoffof0. ii“Alles”—2014/5/8—11:19—page13—#33iiiiii2.3.AdditionalTopics13Intheexhaustedcasegame,thepayoffsforA,B,C,D,Eforthesamevotingpatternare2,1,2,0,1.KnightDstilldoesn'tpreferC,butisnowjustcontentthatsomebodyhasbeenelected.ThatoutcomeisaNashequilibriuminthisversionofthegame.Nobodywould,giventhevotingoftheothers,reconsiderandvotedifferently.KnightsAandDarestillnothappy,buttheycannotunilaterallychangethis.LetmeshowhowwecouldsearchforNashequilibriainthe“exhaustedknight”version.Theideaistostartwithanyoutcome,denedbyasetofchoicesoftheplayers.Ifallplayersareplayingabestresponsetotheotherplayers'moves,wehaveaNashequilibrium.Otherwise,atleastoneplayerdoesnotplayabestresponseyet—weletthisplayerreconsiderandplayabestresponse.Thenweevaluatetheoutcomeagain.EitherwehaveaNashequilibriumnow,orwestillhaveaplayernotplayingthebestresponsetotheotherplayers'moves.Wecontinue,untilwegetaNashequilibrium.Lookattheoutcomewhereeverybodyvotesforhimselfrst.Thiswouldgiveatieandeveryonewouldpreferifhissecondchoicewouldbeelected.So,let'ssayDreconsidersandvotesforBinsteadofforhimself.ThenBwouldbeelected.BandEhavebetterresponses;AcouldvoteforEinsteadforhimselftogetatieandavoidtheelectionofB.NowB,C,andEhavebetterresponses.Let'sassumeBplayshisbestresponseEtotheother'smoves.ThisvotingpatternEECBEturnsouttobeaNashequilibrium.TheprocesscanbesimulatedintheExhaustedKnightsapplet.Initiallyeverybodyvoteshisrstpreference.ChangeD'svotetoB,thenA'svotetoE,thenB'svotetoE.Theprocessdoesnotalwaysterminate;conrmthefollowingintheapplet.Westartwitheveryonevotingforhimself.ThenAchoosesabestresponseandvotesforD.ThenBchoosesabestresponseandvotesforC.AfterthatDreconsidersandvotesforB,thenBreconsidersagain,votingforhimselfagain,andDreconsidersagain,votingforhimselfagain.Afterthiswehaveanoutcomethatwehaddiscussedearlier(votingpatternDBCDE)andtheprocesscouldcontinuethesamewayforever.HistoricalRemarkInthePh.D.thesishewroteatPrincetonUniversityin1950themathematicianJohnForbesNashJr.denedtheequilibriumconceptwhichisnamedafterhim.Laterhedidextraordinaryworkinotherareasofmathematics.Around1959hebecameill,sufferingfromparanoidschizophreniathroughoutthe60sandthe70s.Surprisinglyherecoveredinthe80s.ThereisnoNobelprizeinmathematics,butin1994,withReinhardSeltenandJohnHarsanyi,NashwasawardedtheNobelprizeineconomics(tobemoreprecise,theNobelMemorialPrizeinEconomicSciences).Theawardwasgivenforhisearlyworkingametheory,includinghisdenitionofNashequilibriaandtheexistencetheoremforthem.ThestoryofhislifehasbeentoldinthebookABeautifulMindbySylviaNasar[N1998],whichin2002wasmadeintoanOscar-winningmoviewiththesametitle.2.3AdditionalTopics2.3.1BestResponseDigraphsFora2-playergame,thebestresponseinformationcanbedisplayedinagraph.Thebipartitebestresponsedigraphfortwo-playergamesisdenedasfollows:foreverymoveofAnnwedrawawhitecircleandforeverymoveofBethwedrawablackcircle.Thecirclesarecalledtheverticesofthedigraph.Fromeverywhitevertexwedrawanarrow,anarc,towardsblackverticesthatarebestresponsestothecorrespondingmoveofAnn.Inthesameway,arcsaredrawnfromblackverticestowardsbestresponsewhitevertices.ForExample5,thebestresponsedigraphisshowninFigure2.1. ii“Alles”—2014/5/8—11:19—page14—#34iiiiii142.Theory2:SimultaneousGamesFigure2.1.ThebestresponsedigraphforExample5CondensedBestResponseDigraphsforSymmetricGamesInsymmetricgames,likeADVERTISINGandTWOBARS,itsufcestodisplayacondensedversionofthebestresponsedigraph.Foreverymove(ofeitherplayer—thegameissymmetric,thereforebothplayershavethesamemovesasoptions)wedrawonevertex,andwedrawanarcfrommoveXtomoveYifBeth'sYisabestresponsetoAnn'sX(andthereforealsoAnn'smoveYisabestresponsetoBeth'smoveX).SeeFigures2.2and2.3forthebestresponsedigraphandthecondensedbestresponsedigraphfortheTWOBARSexample.Wemayhavecurvedarcsinthecondensedversion(inourexamplefromvertex4toitself)whenoneofAnn'smove(inourexamplemove4)isthebestresponsetothecorrespondingmove(move4)ofBeth.Figure2.3.ThecondensedbestresponsedigraphforthesymmetricTWOBARSgameFortwo-playergames,Nashequilibriacanberecognizedfromthebestresponsedigraph.Anypairofmoveswitharcsbetweenthem,onebeingthebestresponseoftheother,isaNashequilibrium.Forsymmetric2-persongames,inthecondensedbestresponsedigraphanypairofarcsbetweentwovertices,oranyloopstartingatavertexandpointingtoitselfrepresentsapureNashequilibrium.Thosestemmingfromloopsaresymmetricinsofarasbothplayersusethesamemoveinthem.InsymmetricgamestheymayseemmorenaturalthanasymmetricNashequilibria.2.3.22-PlayerZero-sumSymmetricGamesFortheclassofgameswiththesethreeattributes,Nashequilibria,iftheyexist,canbespottedeasily,andthemaximinpointofviewisthesameastheNashequilibriumview.TheoremIneverysymmetriczero-sumsimultaneousgame,1.everypureNashequilibriumhaszeropayoffforbothplayers,and2.everymaximinmoveofAnnwithsecuritylevel0versusanymaximinmoveofBethwithsecuritylevel0formsaNashequilibrium. ii“Alles”—2014/5/8—11:19—page15—#35iiiiii2.3.AdditionalTopics15Ourrsttheorem!Whatisatheoremanyway?Sofarthischapterhascontainedmostlydenitions,examples,andfactsaboutexamples,suchasthefactthatTWOBARShasoneNashequilibrium.Theo-remsarealsofacts,notaboutsingleconcreteexamplesbutaboutgeneralabstractmathematicalobjectslikesimultaneousgames.Wewanttoprovideaproofforthistheorem.Althoughproofscanbecomplicated,theyjustprovidethereasonswhythetheoremistrue.Youcanacceptthetruthofatheorembasedontheauthorityoftheauthororteacher,soitisallrightifyouskipthe(veryfew)proofsinthisbookonrstreading.Butifyouwanttounderstandamathematicalarea,youalsohavetounderstandthereasoningbehindtheproofs,atleasttosomeextent.Proof1.Lookatanoutcomewhereoneplayer,sayAnn,hasapayoffoflessthanzero.ThenthemovechosenbyAnncouldnotbeherbestresponseforthemovechosenbyBeth,sinceshecanalwaysget0bychoosingthesamemoveasBeth.2.Ann'sminimumpayoffineachrowcannotexceed0,since,ifbothplayerschoosethesameoption,bothhaveapayoffof0.Thereforethesecuritylevelcannotbelargerthan0.IfAnn'sminimumpayoffinarowislessthan0,theneachofBeth'sbestresponsestothemoveofAnncorrespondingtotherowcarriesapayoffofmorethan0forBeth,thereforethismoveofAnncannotbepartofaNashequilibriumby(1).Therefore,ifthesecuritylevelislessthan0,therearenopureNashequilibria.Ifthesecuritylevel(forbothplayers,itisasymmetricgame)equals0,lookatanymaximinmoveforAnnandanymaximinmoveforBeth.ThenAnn'spayoffinthismovecombinationisatleast0,andBeth'spayoffisatleast0.Sincethegameiszero-sum,bothpayoffsmustbeequalto0.Theneachmoveisthebestresponsetotheothermove,andthemovepairformsapureNashequilibrium.Itfollowsthata2-playerzero-sumsymmetricgamehasnopureNashequilibriaprovidedthesecuritylevelislessthan0.AnexampleisROCK-SCISSORS-PAPER.Onefeatureusedintheanalysisofsimultaneousgamesisstillmissing.ItisthetopicofmixingmovesandisdiscussedinChapter27.Exercises1.a)WritethematricesoftheSIMULTANEOUSLEGISLATORSVOTEgameinthevariantwhereeachofthethreevotershasalsotheoptiontoabstain.Theraisepassesonlyifmoreagreethanvotingagainst.Thelossoffacebyabstainingisrelativelysmall,only$200.b)Solvethegame,usingtheapproachesdiscussedabove.2.Considerthefollowingtwo-playergame.LMRU1,13,42,1M2,42,58,1D3,30,40,9Findthemaximinmovesforbothplayers.Whichmovesaredominated?FindthebimatrixobtainedbyIESD.FindthebimatrixobtainedbyIEWD. ii“Alles”—2014/5/8—11:19—page16—#36iiiiii162.Theory2:SimultaneousGamesMarkallbestresponses.ArethereanyNashequilibria?3.Analyzethefollowingtwo-personzero-sumgamesformaximinmoves,domination,bestresponses,andNashequilibria:a)LRU12D34b)LRU12D43c)LRU13D24d)LRU13D42e)LRU14D23f)LRU14D324.Consideratwo-personvariantoftheGUESSTHEAVERAGEgame:AnnandBethsimultaneouslysubmitanumber,1,2,3,or4.Theplayerwhosenumberisclosestto2/3oftheaverageofbothnumbersgets$1.Createthepayoffbimatrix.DecidewhetherthegamehasaNashequilibrium.5.IntheTWOBARSexamplealackoftouristsincreasescompetition.Assumethenumberofnativesis4000.Forwhichnumberoftouristswouldbothbarschoose$4asthepriceforabeer?Forwhichtouristnumbersis$2possible,andforwhichtouristnumbersis$5possible?6.Writethepayoffbimatrixofthefollowinggame.Findmaximinmoves,domination,bestresponses,andpureNashequilibria.SCHEDULINGADINNERPARTY:AnnandBetharenotonspeakingterms,buthavealotofcommonfriends.Bothwanttoinvitethemtoadinnerpartythisweekend,eitherFridayorSaturdayevening.BothslightlypreferSaturday.Ifbothsetthepartyatthesametime,thiswillbeconsideredadisasterwithapayoffof10forboth.IfoneplansthepartyonFridayandtheotheronSaturday,theonehavingtheSaturdaypartygetsapayoffof5,andtheotherof4.7.Analyzethefollowinggame.Createpayoffbimatricesconsistentwiththeinformationgiven.Explainyourchoices.Thenndthemaximinmoves,domination,andallpureNashequilibria.SELECTINGCLASS:Adam,Bill,andCindyareregisteringforaforeignlanguageclassindependentlyandsimultaneously.TheavailableclassesareITA100andFRE100.Theydnotcaremuchwhich,buttheycarewithwhomtheysharetheclass.BillandCindywanttobeinthesameclass,butwanttoavoidAdam.AdamwantstobeinthesameclassasBillorCindy,orevenbetter,both.8.DEADLOCK:Twoplayersplayasymmetricgamewhereeachcaneithercooperateordefect.Iftheycooperate,bothgetanpayoffof1.Iftheydefect,bothgetapayoffof2.Ifonecooperatesbuttheotherdefects,theonecooperatinggetsapayoffof0,andtheonedefectingapayoffof3.Drawthebimatrixofthegame.Findthemaximinmoves,possibledomination,bestresponses,andndallpureNashequilibria.9.STAGHUNT:Twoplayersplayasymmetricgamewhereeachcanhunteitherstagorhare.Ifbothhuntstag,bothgetanpayoffof3.Ifbothhunthare,bothgetapayoffof1.Ifonehuntsstagandtheotherhare,thestaghuntergetsapayoffof0,andtheharehunterapayoffof2.Drawthebimatrixofthegame.Findthemaximinmoves,possibledomination,bestresponses,andndallpureNashequilibria.10.CHICKEN:Twoplayersplayasymmetricgamewhereeachcaneitherplaydoveorhawk.Ifbothplaydove,bothgetanpayoffof2.Ifbothplayhawk,bothgetapayoffof0.Ifoneplaysdoveandtheotherhawk,theoneplayingdovegetsapayoffof1,andtheotheroneapayoffof3.Drawthebimatrixofthegame.Findthemaximinmoves,possibledomination,bestresponses,andndallpureNashequilibria. ii“Alles”—2014/5/8—11:19—page17—#37iiiiii2.3.AdditionalTopics1711.BULLY:Twoplayersplaythefollowinggamecooperatedefectcooperate2,11,3defect3,00,2(compare[P1993]).Findthemaximinmoves,possibledomination,bestresponses,andndallpureNashequilibria.12.TwocarsaremeetingatanintersectionandwanttoproceedasindicatedbythearrowsinFigure2.4.Eachplayercanproceedormove.Ifbothproceed,thereisanaccident.Awouldhaveapayoffof100inthiscase,andBapayoffof1000(sinceBwouldbemaderesponsiblefortheaccident,sinceAhastherightofway).Ifoneyieldsandtheotherproceeds,theoneyieldinghasapayoffof5,andtheotheroneof5.Ifbothyield,ittakesalittlelongeruntiltheycanproceed,sobothhaveapayoffof10.Analyzethissimultaneousgame,drawthepayoffbimatrix,andndpureNashequilibria.Figure2.4.Twocarsatacrossing13.ThreecarsaremeetingatanintersectionandwanttoproceedasindicatedbythearrowsinFigure2.5.Eachplayercanproceedormove.Iftwowithintersectingpathsproceed,thereisanaccident.Theonehavingtherightofwayhasapayoffof100inthiscase,theotheroneapayoffof1000.Ifacarproceedswithoutcausinganaccident,thepayoffforthatcaris5.Ifacaryieldsandalltheothersintersectingitspathproceed,theyieldingcarhasapayoffof5.Ifacaryieldsandaconictingpathcaraswell,ittakesalittlelongeruntiltheycanproceed,sobothhaveapayoffof10.Analyzethissimultaneousgame,drawthepayoffbimatrices,andndallpureNashequilibria.Figure2.5.Threecarsatacrossing14.SolvetheSIMULTANEOUSULTIMATUMGAME.Displaythepayoffbimatrix,andinvestigatemax-iminmoves,domination,bestresponses,andwhetherthereareanyequilibria.15.AnalyzeaversionoftheBATTLEOFTHESEXESexamplewhereonepartnerhashighloveandtheotherlowlove.Forthehighlovepartner,beingtogetherwiththepartnerismoreimportantthanbeingatthepreferredlocation,whereasforthelowlovepartneritistheopposite.ArethereNashequilibria? ii“Alles”—2014/5/8—11:19—page18—#38iiiiii182.Theory2:SimultaneousGames16.Assumethatasimultaneoustwo-playergamehasthebestresponsedigraphshowninFigure2.6.Figure2.6.AbestresponsedigraphDisplayapossiblepayoffbimatrix.Canyoundazero-sumpayoffbimatrixgeneratingthisbestre-sponsedigraph?17.Inthe5KNIGHTSgamedescribedinExample8withpreferencesasdescribedthere,determinewhetherthevotingpatternECEDE(AvotesforE,BvotesforC,etc.)formsaNashequilibriumintheearlycasegameorintheexhaustedcasegame.18.Inthe5KNIGHTSgamedescribedinExample8withpreferencesasdescribedthere,determinewhetherthevotingpatternACEBE(AvotesforA,BvotesforC,etc.)formsaNashequilibriumintheearlycasegameorintheexhaustedcasegame.19.Usethe5KNIGHTSapplettondaNashequilibriumfortheearlyversionofthe5KNIGHTSgame.Project1ReactingfastorslowAssumetheveplayersofthe5KNIGHTSgamearenegotiatingwhattovotebeforeactuallyvoting.Theystartwiththeobviousproposalofeverybodyvotingforhisrstchoice.Ifsomebody'smoveisnotthebestresponsetotheotherplayers'moveshechangestheproposal,proposinghisbestresponseashismove.ThisisrepeateduntilaNashequilibriumisfound.Theprocessisnotunique,sinceiftwoormoreplayerswanttoreconsider,onlyonewillmodifytheproposalatatime.Discusswhetheritisbetterinsuchanegotiationtoalwaysmodifyearly,ortowaitandseewhethertheotherschangetheproposalrst.Simulatetheprocessfordifferentpreferencesinthe5KNIGHTSRANDOMappletwiththeassumptionthatAalwaysreactsfasterthanB,BalwaysreactsfasterthanC,andsoon.Dothisatleast30times,andkeeptrackhowofteneachoneoftheplayersiselectedintheresultingNashequilibrium(whichyouhopefullyget—theremayalsobecycliccasesasdiscussedinExample8ofChapter27). 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ii“Alles”—2014/5/8—11:19—page284—#304iiiiiiIndex1-CANDIDATEQUIZSHOW,1232FINGERS,562-AUCTION,693-AUCTION,695CARDINALS,455ENVELOPES,1245KNIGHTS,125PIRATES,685PIRATESwithrandomselection,1268ROUNDSBLACKORWHITE,147ABeautifulMind,212ABABA,112ABABAB,70ABBAAB,70ABBABA,70,112action,1acyclicdigraph,57,160adjacent,25,34Akerlof,GeorgeA.,278alternatives,160altruism,60arc,13arcs,54Aumann,Robert,157,278backwardinductionforperfectinformationandnorandomness,59forperfectinformationandrandomness,122Banzhaf-PenroseIndex,52BARGAINING,65BATTLEOFTHESEXES,10behavioralstrategy,231bestresponse,8bestresponsedigraph,13Borel,Emile,63bottom-upanalysis,75branchsubgame,164Brown'sctitiousplay,200Brown,GeorgeW.,156C&C,113Cardcounting,135cardinalscale,90CENTIPEDEGAME,60CHICKEN,81CHOCOLATESINTHEBOX,68COLONELBLOTTO,157completeinformation,2conditionalprobability,91cooperativegame,2core,256CRASHROCK-SCISSORS-PAPER,201CREATINGHOLES,66cycle,33cyclicdigraph,57Dantzig,George,156,204,213digraphacyclic,160probability,93DMA6*SOCCER,109DMA7*SOCCER,109Dresher,Melvin,158DUEL,149,152two-bullet,149,152edge,34ELECTION,50,157empiricalprobability,87endpositions,53endvertex,33,160envy-free,111equilibriumNash,10event,86Excel,42cells,42inputcells,42outputcells,42experiments,86284 ii“Alles”—2014/5/8—11:19—page285—#305iiiiiiIndex285extensiveform,54,159fair,80Ferguson,Chris,138ctitiousplay,200FINALEXAM,98nalvertex,160rstmoveradvantage,80Flood,Merrill,158gaindigraph,72gainvalue,72Gale,David,279game,1nite,56symmetric,6zero-sum,5gamedigraph,54,159gametree,55,160games1-CANDIDATEQUIZSHOW,1232FINGERS,562-AUCTION,693-AUCTION,695Cardinals,455ENVELOPES,1245Knights,125PIRATES,685PIRATESwithrandomselection,1268ROUNDSBLACKORWHITE,147ABABA,112ABABAB,112ABBABA,112BARGAINING,65C&C,113CENTIPEDEGAME,60CHICKEN,81CHOCOLATESINTHEBOX,68COLONELBLOTTO,157CRASHROCK-SCISSORS-PAPER,201CREATINGHOLES,66DMA6*SOCCER,109DMA7*SOCCER,109DUEL,149ELECTION,50,157FINALEXAM,98iteratedCOLONELBLOTTO,211JOBINTERVIEWS,127LISA'SGAME,69MATCHINGCHAIRS,65MINIBLACKJACK,142MINIBLACKJACK(a;b),139NIM(6),53OH-NO,124OH-NO6,128PolyominoRECTHESQUAREwithRandom-ness,125PolyominoRECTHESQUAREwithrandom-ness,128PRISONER'SDILEMMA,158R&E2,113R&E4,113RANDOMNIM(n;p),121RECTHESQUARE,66ROCK-SCISSORS-PAPER,81ROCK-SCISSORS-PAPER2,202SEANATERACEII,54SENATERACE,96SEQUENTIALLEGISLATORSVOTE,57SHUBIKAUCTION,77,169SimultaneousQuatro-Uno,44symmetric,80TAKESOME,66,69TENNIS:,97ULTIMATUMGAME,60WHO'SNEXT,67xROUNDSRED,GREEN,ORBLUE,148graph,34greedystrategy,70guaranteedpayoffs,62Harsanyi,John,181,277Hurwicz,Leonid,278IESD,7IEWD,8inefciency,111inequity,111informationset,159intervalscale,90isomorphic,157iteratedCOLONELBLOTTO,211JOBINTERVIEWS,127JuliaRobinson'stheorem,203 ii“Alles”—2014/5/8—11:19—page286—#306iiiiii286IndexKellycriterion,136,137KellyJr.,J.L.,135Kimmel,Emmanuel,135KuhnPoker,186Kuhn,H.W.,185Kuhn,HaroldWilliam,279lawoflargenumbers,87LeHerLeHer*(S,r),178linearprogram,203LISA'SGAME,69Marcum,Jess,135Maskin,EricS.,278MATCHINGCHAIRS,65maximalouterplanargraph,33maximinmove,6maximinstrategy,6maximinvalue,6medianofagraph,30MinSat,112MINIBLACKJACK,142MINIBLACKJACK(a;b),139Mirrlees,JamesA.,278mixedmaximinstrategy,202mixedNashequilibrium,203mixedsecuritylevel,202mixedstrategy,199MOP,33Morgenstern,Oskar,185move,1moves,160Myerson,RogerB.,278Nashequilibriummixed,203Pareto-optimal,11pure,10subgameperfect,181Nash'sTheorem,204Nash,John,13,135,156,181,212,277neighbor,34netodds,136NIM(6),53NobelMemorialPrizeinEconomics,13,157Nobel,Alfred,277nominalscale,90normalform,4,5,177OH-NO,124OH-NO6,128options,160ordinalscale,90outcome,4outcomes,1Pareto-dominated,11,111Pareto-optimal,11payoff,1perfectinformation,2perfectrecall,233play,1PolyominoRECTHESQUAREwithRandomness,125PolyominoRECTHESQUAREwithrandomness,128position,1,53PRISONER'SDILEMMA,10,158probability,87empirical,87theoretical,87probabilitydigraph,93probabilitytree,91projects5Cardinals,455ENVELOPES,1275Knights,12JOBINTERVIEWS,127OH-NO,128PolyominoRECTHESQUAREwithrandom-ness,128RestaurantLocationGames,44SimultaneousQuatro-Uno,44proof,15pureNashequilibrium,10purestrategy,176R&E2,113R&E4,113randommoves,2RANDOMNIM(n;p),121randomvariable,89randomvertex,121ratioscale,90RECTHESQUARE,66 ii“Alles”—2014/5/8—11:19—page287—#307iiiiiiIndex287reducedpurestrategy,177relativefrequency,87relativevalues,111ROCK-SCISSORS-PAPER,81ROCK-SCISSORS-PAPER2,202Roth,AlvinE.,278satisfaction,111scalecardinal,90interval,90nominal,90ordinal,90ratio,90Schelling,Thomas,157,278secondmoveradvantage,80securitylevel,6,62securitylevelanalysis,62securitystrategy,62Selten,Reinhard,181,277SENATERACE,96SENATERACEII,54sequentialgame,2,53SEQUENTIALLEGISLATORSVOTE,57setposition,72Shapleyvalue,256Shapley,Lloyd,278Shapley,LloydStowell,135,156SHUBIKAUCTION,77,169Shubik,Martin,135,279simpleevent,86simultaneousgame,2SimultaneousQuatro-Uno,44Spence,A.Michael,278spreadsheetprogram,42stablecoalition,255standarddeviation,115startposition,53startvertex,160Stiglitz,JosephE.,278strategicform,177strategy,1pure,176reduced,177strictdomination,7subgame,164subgameperfect,181support,200swingvoter,52symmetricgames,80TAKESOME,66,69TENNIS,97terminalvertex,160theorem,15theoreticalprobability,87Thorp,EdwardO.,135ULTIMATUMGAME,60uniqueresponse,84Uston,Ken,138utilities,60utilityfunction,136valueofatwo-playerzero-sumgame,202variance,116vertex,13,34vertices,54Vickrey,William,278VNMPoker,185vonNeumann'sTheorem,203vonNeumann,John,63,156,185,212weakdomination,8WHO'SNEXT,67Williams,JohnDavid,157winningcoalition,52xROUNDSRED,GREEN,ORBLUE,148Zermelo'sTheorem,59Zermelo,Ernst,59zero-sum,5zero-sumgame,2 through EXAMPLESPRISNER Game CLASSROOM RESOURCE MATERIALS