Game Theory: A Playful Introduction PDF
Preview Game Theory: A Playful Introduction
STUDENT MATHEMATICAL LIBRARY Volume 80 Game Theory A Playful Introduction Matt DeVos Deborah A. Kent Game Theory A Playful Introduction https://doi.org/10.1090//stml/080 STUDENT MATHEMATICAL LIBRARY Volume 80 Game Theory A Playful Introduction Matt DeVos Deborah A. Kent Providence, Rhode Island EditorialBoard SatyanL.Devadoss JohnStillwell(Chair) EricaFlapan SergeTabachnikov 2010MathematicsSubjectClassification.Primary91-01,91A46,91A06,91B06. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/stml-80 LibraryofCongressCataloging-in-PublicationData Names:DeVos,MatthewJared,1974-jKent,DeborahA.,1978- Title:Gametheory:aplayfulintroduction/MatthewDeVos,DeborahA.Kent. Description:Providence,RhodeIsland:AmericanMathematicalSociety,[2016]jSeries: Studentmathematicallibrary; volume80jIncludesbibliographicalreferencesand index. Identifiers:LCCN2016035452jISBN9781470422103(alk.paper) Subjects: LCSH:Gametheory–Textbooks. jCombinatorialanalysis–Textbooks. jAMS: Game theory, economics, social and behavioral sciences – Instructional exposition (textbooks,tutorialpapers,etc.).mscjGametheory,economics,socialandbehavioral sciences–Gametheory–Combinatorialgames.mscjGametheory,economics,social andbehavioralsciences–Gametheory–n-persongames,n > 2. mscjGamethe- ory,economics,socialandbehavioralsciences–Mathematicaleconomics–Decision theory.msc Classification: LCC QA269 .D45 2016 j DDC 519.3–dc23 LC record available at https://lccn.loc.gov/2016035452 Copyingandreprinting.Individualreadersofthispublication,andnonprofitlibrariesactingfor them,arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteachingor research.Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,providedthe customaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationis permittedonlyunderlicensefromtheAmericanMathematicalSociety.Permissionstoreuseportions ofAMSpublicationcontentarehandledbyCopyrightClearanceCenter’sRightsLink®service. For moreinformation,pleasevisit:http://www.ams.org/rightslink. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).Copy- rightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthefirstpage ofeacharticlewithinproceedingsvolumes. ©2016bytheAmericanMathematicalSociety.Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. ⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 212019181716 Dedicatedtomyfamily,MD. ForMom,inmemorywithlove,DK. Contents Preface xi Chapter1. CombinatorialGames 1 §1.1. GameTrees 3 §1.2. Zermelo’sTheorem 9 §1.3. Strategy 14 Exercises 19 Chapter2. Normal-PlayGames 25 §2.1. PositionsandTheirTypes 27 §2.2. SumsofPositions 30 §2.3. Equivalence 36 Exercises 41 Chapter3. ImpartialGames 45 §3.1. Nim 46 §3.2. TheSprague-GrundyTheorem 52 §3.3. ApplyingtheMEXPrinciple 54 Exercises 59 vii viii Contents Chapter4. HackenbushandPartizanGames 63 §4.1. Hackenbush 64 §4.2. DyadicNumbersandPositions 71 §4.3. TheSimplicityPrinciple 77 Exercises 83 Chapter5. Zero-SumMatrixGames 89 §5.1. Dominance 91 §5.2. MixedStrategies 95 §5.3. VonNeumannSolutions 100 Exercises 104 Chapter6. VonNeumann’sMinimaxTheorem 111 §6.1. EquatingtheOpponent’sResults 113 §6.2. Two-DimensionalGames 118 §6.3. ProofoftheMinimaxTheorem 123 Exercises 128 Chapter7. GeneralGames 133 §7.1. Utility 135 §7.2. MatrixGames 139 §7.3. GameTrees 145 §7.4. Treesvs.Matrices 150 Exercises 155 Chapter8. NashEquilibriumandApplications 161 §8.1. NashEquilibrium 162 §8.2. EvolutionaryBiology 169 §8.3. CournotDuopoly 176 Exercises 182 Chapter9. Nash’sEquilibriumTheorem 187 §9.1. Sperner’sLemma 189 §9.2. Brouwer’sFixedPointTheorem 192 Contents ix §9.3. StrategySpaces 198 §9.4. NashFlowandtheProof 202 Exercises 208 Chapter10. Cooperation 213 §10.1. TheNegotiationSet 214 §10.2. NashArbitration 221 §10.3. RepeatedGamesandtheFolkTheorem 228 Exercises 238 Chapter11. 𝑛-PlayerGames 245 §11.1. MatrixGames 247 §11.2. Coalitions 251 §11.3. ShapleyValue 260 Exercises 270 Chapter12. PreferencesandSociety 275 §12.1. FairDivision 277 §12.2. StableMarriages 285 §12.3. Arrow’sImpossibilityTheorem 290 Exercises 298 AppendixA. OnGamesandNumbers 301 AppendixB. LinearProgramming 309 BasicTheory 310 AConnectiontoGameTheory 313 LPDuality 317 AppendixC. NashEquilibriuminHighDimensions 323 GameBoards 331 Bibliography 335 IndexofGames 339 Index 341
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