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 FernandoGouveâ,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.Sańchez 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

Game Theory: Through Examples PDF

308 Pages·2014·4.666 MB·English

by Erich Prisner

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