Mathematical game theory and applications

431 Pages·2014·2.31 MB·English

by  Vladimir Mazalov

Mathematical
Game Theory
and Applications
Vladimir Mazalov

Mathematical Game Theory
and Applications

Mathematical Game Theory
and Applications
Vladimir Mazalov
ResearchDirectoroftheInstitute ofAppliedMathematicalResearch,
KareliaResearchCenterofRussianAcademyofSciences,Russia
Thiseditionfirstpublished2014
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LibraryofCongressCataloging-in-PublicationData
Mazalov,V.V.(VladimirViktorovich),author.
Mathematicalgametheoryandapplications/VladimirMazalov.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-89962-5(hardback)
1.Gametheory. I.Title.
QA269.M4152014
519.3–dc23
2014019649
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-1-118-89962-5
Setin10/12ptTimesbyAptaraInc.,NewDelhi,India.
1 2014
Contents
Preface xi
Introduction xiii
1 Strategic-FormTwo-PlayerGames 1
Introduction 1
1.1 TheCournotDuopoly 2
1.2 ContinuousImprovementProcedure 3
1.3 TheBertrandDuopoly 4
1.4 TheHotellingDuopoly 5
1.5 TheHotellingDuopolyin2DSpace 6
1.6 TheStackelbergDuopoly 8
1.7 ConvexGames 9
1.8 SomeExamplesofBimatrixGames 12
1.9 Randomization 13
1.10 Games2×2 16
1.11 Games2×nandm×2 18
1.12 TheHotellingDuopolyin2DSpacewithNon-UniformDistribution
ofBuyers 20
1.13 LocationProblemin2DSpace 25
Exercises 26
2 Zero-SumGames 28
Introduction 28
2.1 MinimaxandMaximin 29
2.2 Randomization 31
2.3 GameswithDiscontinuousPayoffFunctions 34
2.4 Convex-ConcaveandLinear-ConvexGames 37
2.5 ConvexGames 39
2.6 ArbitrationProcedures 42
2.7 Two-PointDiscreteArbitrationProcedures 48
2.8 Three-PointDiscreteArbitrationProcedureswithIntervalConstraint 53
vi CONTENTS
2.9 GeneralDiscreteArbitrationProcedures 56
Exercises 62
3 Non-CooperativeStrategic-Formn-PlayerGames 64
Introduction 64
3.1 ConvexGames.TheCournotOligopoly 65
3.2 PolymatrixGames 66
3.3 PotentialGames 69
3.4 CongestionGames 73
3.5 Player-SpecificCongestionGames 75
3.6 Auctions 78
3.7 WarsofAttrition 82
3.8 Duels,Truels,andOtherShootingAccuracyContests 85
3.9 PredictionGames 88
Exercises 93
4 Extensive-Formn-PlayerGames 96
Introduction 96
4.1 EquilibriuminGameswithCompleteInformation 97
4.2 IndifferentEquilibrium 99
4.3 GameswithIncompleteInformation 101
4.4 TotalMemoryGames 105
Exercises 108
5 ParlorGamesandSportGames 111
Introduction 111
5.1 Poker.AGame-TheoreticModel 112
5.1.1 OptimalStrategies 113
5.1.2 SomeFeaturesofOptimalBehaviorinPoker 116
5.2 ThePokerModelwithVariableBets 118
5.2.1 ThePokerModelwithTwoBets 118
5.2.2 ThePokerModelwithnBets 122
5.2.3 TheAsymptoticPropertiesofStrategiesinthePokerModelwith
VariableBets 127
5.3 Preference.AGame-TheoreticModel 129
5.3.1 StrategiesandPayoffFunction 130
5.3.2 EquilibriumintheCaseof B−A ≤ 3A−B 132
B+C 2(A+C)
5.3.3 EquilibriumintheCaseof 3A−B < B−A 134
2(A+C) B+C
5.3.4 SomeFeaturesofOptimalBehaviorinPreference 136
5.4 ThePreferenceModelwithCardsPlay 136
5.4.1 ThePreferenceModelwithSimultaneousMoves 137
5.4.2 ThePreferenceModelwithSequentialMoves 139
5.5 Twenty-One.AGame-TheoreticModel 145
5.5.1 StrategiesandPayoffFunctions 145
5.6 Soccer.AGame-TheoreticModelofResourceAllocation 147
Exercises 152
CONTENTS vii
6 NegotiationModels 155
Introduction 155
6.1 ModelsofResourceAllocation 155
6.1.1 CakeCutting 155
6.1.2 PrinciplesofFairCakeCutting 157
6.1.3 CakeCuttingwithSubjectiveEstimatesbyPlayers 158
6.1.4 FairEqualNegotiations 160
6.1.5 Strategy-Proofness 161
6.1.6 SolutionwiththeAbsenceofEnvy 161
6.1.7 SequentialNegotiations 163
6.2 NegotiationsofTimeandPlaceofaMeeting 166
6.2.1 SequentialNegotiationsofTwoPlayers 166
6.2.2 ThreePlayers 168
6.2.3 SequentialNegotiations.TheGeneralCase 170
6.3 StochasticDesignintheCakeCuttingProblem 171
6.3.1 TheCakeCuttingProblemwithThreePlayers 172
6.3.2 NegotiationsofThreePlayerswithNon-UniformDistribution 176
6.3.3 NegotiationsofnPlayers 178
6.3.4 NegotiationsofnPlayers.CompleteConsent 181
6.4 ModelsofTournaments 182
6.4.1 AGame-TheoreticModelofTournamentOrganization 182
6.4.2 TournamentforTwoProjectswiththeGaussianDistribution 184
6.4.3 TheCorrelationEffect 186
6.4.4 TheModelofaTournamentwithThreePlayersand
Non-ZeroSum 187
6.5 BargainingModelswithIncompleteInformation 190
6.5.1 TransactionswithIncompleteInformation 190
6.5.2 HonestNegotiationsinConclusionofTransactions 193
6.5.3 TransactionswithUnequalForcesofPlayers 195
6.5.4 The“Offer-Counteroffer”TransactionModel 196
6.5.5 TheCorrelationEffect 197
6.5.6 TransactionswithNon-UniformDistributionof
ReservationPrices 199
6.5.7 TransactionswithNon-LinearStrategies 202
6.5.8 TransactionswithFixedPrices 207
6.5.9 EquilibriumAmongn-ThresholdStrategies 210
6.5.10 Two-StageTransactionswithArbitrator 218
6.6 ReputationinNegotiations 221
6.6.1 TheNotionofConsensusinNegotiations 221
6.6.2 TheMatrixFormofDynamicsintheReputationModel 222
6.6.3 InformationWarfare 223
6.6.4 TheInfluenceofReputationinArbitrationCommittee.
ConventionalArbitration 224
6.6.5 TheInfluenceofReputationinArbitrationCommittee.
Final-OfferArbitration 225
6.6.6 TheInfluenceofReputationonTournamentResults 226
Exercises 228
viii CONTENTS
7 OptimalStoppingGames 230
Introduction 230
7.1 OptimalStoppingGame:TheCaseofTwoObservations 231
7.2 OptimalStoppingGame:TheCaseofIndependentObservations 234
7.3 TheGameΓ (G)UnderN ≥3 237
N
7.4 OptimalStoppingGamewithRandomWalks 241
7.4.1 SpectraofStrategies:SomeProperties 243
7.4.2 EquilibriumConstruction 245
7.5 BestChoiceGames 250
7.6 BestChoiceGamewithStoppingBeforeOpponent 254
7.7 BestChoiceGamewithRankCriterion.Lottery 259
7.8 BestChoiceGamewithRankCriterion.Voting 264
7.8.1 SolutionintheCaseofThreePlayers 265
7.8.2 SolutionintheCaseofmPlayers 268
7.9 BestMutualChoiceGame 269
7.9.1 TheTwo-ShotModelofMutualChoice 270
7.9.2 TheMulti-ShotModelofMutualChoice 272
Exercises 276
8 CooperativeGames 278
Introduction 278
8.1 EquivalenceofCooperativeGames 278
8.2 ImputationsandCore 281
8.2.1 TheCoreoftheJazzBandGame 282
8.2.2 TheCoreoftheGloveMarketGame 283
8.2.3 TheCoreoftheSchedulingGame 284
8.3 BalancedGames 285
8.3.1 TheBalanceConditionforThree-PlayerGames 286
8.4 The𝜏-ValueofaCooperativeGame 286
8.4.1 The𝜏-ValueoftheJazzBandGame 289
8.5 Nucleolus 289
8.5.1 TheNucleolusoftheRoadConstructionGame 291
8.6 TheBankruptcyGame 293
8.7 TheShapleyVector 298
8.7.1 TheShapleyVectorintheRoadConstructionGame 299
8.7.2 Shapley’sAxiomsfortheVector𝜑(v) 300
i
8.8 VotingGames.TheShapley–ShubikPowerIndexandtheBanzhafPower
Index 302
8.8.1 TheShapley–ShubikPowerIndexforInfluenceEvaluationinthe
14thBundestag 305
8.8.2 TheBanzhafPowerIndexforInfluenceEvaluationinthe3rdState
Duma 307
8.8.3 TheHollerPowerIndexandtheDeegan–PackelPowerIndexfor
InfluenceEvaluationintheNationalDiet(1998) 309
8.9 TheMutualInfluenceofPlayers.TheHoede–BakkerIndex 309
Exercises 312