Table Of Content

i i “Alles” — 2014/5/8 — 11:19 — page i — #1 i i Game Theory Through Examples i i i i i i “Alles” — 2014/5/8 — 11:36 — page ii — #2 i i c 2014bytheMathematicalAssociationofAmerica,Inc. (cid:13) ElectroniceditionISBN978-1-61444-115-1 i i i i i i “Alles” — 2014/5/8 — 11:19 — page iii — #3 i i Game Theory Through Examples Erich Prisner Franklin University Switzerland PublishedandDistributedby TheMathematicalAssociationofAmerica i i i i i i “Alles” — 2014/5/8 — 11:19 — page iv — #4 i i CouncilonPublicationsandCommunications FrankFarris,Chair CommitteeonBooks FernandoGouveˆa,Chair ClassroomResourceMaterialsEditorialBoard SusanGStaples,Editor MichaelBardzell JenniferBergner CarenLDiefenderfer ChristopherHallstrom CynthiaJHuffman PaulRKlingsberg BrianLins MaryMorley PhilipPMummert BarbaraEReynolds DarrylYong i i i i i i “Alles” — 2014/5/8 — 11:19 — page v — #5 i i CLASSROOMRESOURCEMATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students— laboratoryexercises,projects,historicalinformation,textbookswithunusualapproachesforpresentingmath- ematicalideas,careerinformation,etc. 101CareersinMathematics,3rdeditioneditedbyAndrewSterrett Archimedes: WhatDidHeDoBesidesCryEureka?,ShermanStein Calculus: An Active Approach withProjects, Stephen Hilbert, Diane DriscollSchwartz, Stan Seltzer, John Maceli,andEricRobinson CalculusMysteriesandThrillers,R.GrantWoods ConjectureandProof,Miklo´sLaczkovich CounterexamplesinCalculus,SergiyKlymchuk CreativeMathematics,H.S.Wall EnvironmentalMathematicsintheClassroom,editedbyB.A.FusaroandP.C.Kenschaft ExcursionsinClassicalAnalysis: PathwaystoAdvancedProblemSolvingandUndergraduateResearch, by HongweiChen ExplorationsinComplexAnalysis,MichaelA.Brilleslyper,MichaelJ.Dorff,Jane M.McDougall,James S. Rolf,LisbethE.Schaubroeck,RichardL.Stankewitz,andKennethStephenson ExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.Weller ExploringAdvancedEuclideanGeometrywithGeoGebra,GerardA.Venema GameTheoryThroughExamples,ErichPrisner GeometryFromAfrica: MathematicalandEducationalExplorations,PaulusGerdes HistoricalModulesfor the Teaching andLearning of Mathematics(CD), edited by VictorKatz and Karen DeeMichalowicz IdentificationNumbersandCheckDigitSchemes,JosephKirtland InterdisciplinaryLivelyApplicationProjects,editedbyChrisArney InverseProblems: ActivitiesforUndergraduates,CharlesW.Groetsch KeepingitR.E.A.L.: ResearchExperiences forAllLearners,CarlaD.MartinandAnthonyTongen LaboratoryExperiencesinGroupTheory,EllenMaycockParker LearnfromtheMasters,FrankSwetz,JohnFauvel,OttoBekken,BengtJohansson,andVictorKatz MathMadeVisual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaandRogerB.Nelsen MathematicsGalore!: TheFirstFiveYearsoftheSt. MarksInstituteofMathematics,JamesTanton MethodsforEuclideanGeometry,OwenByer,FelixLazebnik,andDeirdreL.Smeltzer OrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.Sa´nchez OvalTrackandOtherPermutationPuzzles,JohnO.Kiltinen ParadoxesandSophismsinCalculus,SergiyKlymchukandSusanStaples APrimerofAbstractMathematics,RobertB.Ash ProofsWithoutWords,RogerB.Nelsen ProofsWithoutWordsII,RogerB.Nelsen RediscoveringMathematics:YouDotheMath,ShaiSimonson i i i i i i “Alles” — 2014/5/8 — 11:19 — page vi — #6 i i SheDoesMath!,editedbyMarlaParker SolveThis:MathActivitiesforStudentsandClubs,JamesS.Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, David Williamson,MarilouMendel,JulieTarr,andDeborahYoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson,MarilouMendel,JulieTarr,andDeborahYoklic TeachingStatisticsUsingBaseball,JimAlbert VisualGroupTheory,NathanC.Carter WhichNumbersareReal?,MichaelHenle WritingProjects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, GavinLaRose,ThomasRatliff,andElynRykken MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1MAA FAX:1-301-206-9789 i i i i i i “Alles” — 2014/5/8 — 11:19 — page vii — #7 i i Contents Preface xvi 1 Theory1: Introduction 1 1.1 What’saGame? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Game,Play,Move: SomeDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 ClassificationofGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Theory2: SimultaneousGames 4 2.1 NormalForm—BimatrixDescription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 TwoPlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 TwoPlayers,Zero-sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 ThreeorMorePlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 SymmetricGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 WhichOptiontoChoose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 MaximinMoveandSecurityLevel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 DominatedMoves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3 BestResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 NashEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 AdditionalTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 BestResponseDigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 2-PlayerZero-sumSymmetricGames . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Project1:Reactingfastorslow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Example:SelectingaClass 19 3.1 ThreePlayers,TwoClasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 “Ilikeyouboth” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 DislikingtheRival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.3 Outsider. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 LargerCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Project2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Project3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Project4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 vii i i i i i i “Alles” — 2014/5/8 — 11:19 — page viii — #8 i i viii Contents 4 Example:DoctorLocationGames 25 4.1 DoctorLocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.1 AnExampleGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1.2 No(Pure)NashEquilibrium? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.3 HowGoodaretheNashEquilibriaforthePublic? . . . . . . . . . . . . . . . . . . . 28 4.2 Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 MorethanoneOffice(optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Project5:DoctorlocationonMOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Project6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Project7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Example:RestaurantLocationGames 34 5.1 AFirstGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 ASecondGraph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 ExistenceofPureNashEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 MorethanoneRestaurant(optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 UsingExcel 42 6.1 SpreadsheetProgramslikeExcel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 Two-PersonSimultaneousGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Three-PersonSimultaneousGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Project8:SimultaneousQuatro-Uno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Project9:RestaurantLocationGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Project10:5Knights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Project11:5Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 Example:ElectionI 47 7.1 FirstExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 SecondExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.3 TheGeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.4 ThirdExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.5 TheEightCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.6 VotingPowerIndices(optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8 Theory3: SequentialGamesI:PerfectInformationandnoRandomness 53 8.1 ExtensiveForm: GameTreeandGameDigraph . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.2 AnalyzingtheGame: BackwardInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.2.1 FiniteGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.2.2 TheProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.2.3 Zermelo’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3 AdditionalTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3.1 RealityCheck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3.2 PlayingitSafe—GuaranteedPayoffs . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8.3.3 Two-personZero-sumGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 i i i i i i “Alles” — 2014/5/8 — 11:19 — page ix — #9 i i Contents ix 8.3.4 BreakingTies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.3.5 ExistingGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.3.6 GreedyStrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Project12:TAKESOME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Project13:WHO’sNEXT(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Project14:LISA’sGAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Project15:2-AUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Project16:3-AUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9 Example:DividingAFewItemsI 70 9.1 GreedyStrategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 9.2 BackwardInduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 9.2.1 GameTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 9.2.2 GameDigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 9.2.3 Example: GameDigraphforABBAB . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.3 AnAbbreviatedAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.3.1 WhyitMatters:Complexity(optional) . . . . . . . . . . . . . . . . . . . . . . . . . 74 9.4 Bottom-UpAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 9.5 InterdependenciesbetweentheItems(optional) . . . . . . . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 10 Example:ShubikAuctionI 77 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Project17:SHUBIKAUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 11 Example:SequentialDoctorandRestaurantLocation 80 11.1 GeneralObservationsforSymmetricGames . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 11.2 DoctorLocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 11.3 Constant-SumGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 11.4 RestaurantLocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11.5 NashEquilibriaandFirstMoverAdvantageforSymmetricGames . . . . . . . . . . . . . . . 84 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Project18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Project19:HostileversusFriendlyPlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 12 Theory4: Probability 86 12.1 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 12.2 ComputingProbabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.2.1 EquallyLikelySimpleEvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.2.2 SimpleEventsnotEquallyLikely . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.3 ExpectedValue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 12.4 MultistepExperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 12.4.1 ProbabilityTrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 12.4.2 ConditionalProbabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 12.4.3 ProbabilityDigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 12.5 RandomnessinSimultaneousGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 12.6 CountingwithoutCounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 i i i i

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Game Theory Through Examples is a thorough introduction to elementary game theory, covering finite games with complete information. The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential
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Game Theory Through Examples is a thorough introduction to elementary game theory, covering finite games with complete information. The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential

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