Game Theory Evolving

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收藏 2010-03-18

Preface 17
Suggestions for Instructors 25
1 Game Theory: A Lexicon for Strategic Interaction 33
1.1 Introduction........................ 33
1.2 BigMonkeyandLittleMonkey ............. 33
1.3 TheExtensiveFormGame................ 39
1.4 TheNormalFormGame................. 42
1.5 NashEquilibrium..................... 43
1.6 ReviewingtheTerminology ............... 44
2 Leading from Strength: Playing Dominant Strategies 46
2.1 Introduction........................ 46
2.2 DominantStrategies ................... 46
2.3 Backward Induction: Pruning the Game Tree . .... 47
2.4 EliminatingDominatedStrategies............ 49
2.5 DominatedStrategies:ConceptsandDenitions.... 49
2.6 ThePrisoner’sDilemma ................. 50
2.7 AnArmamentsGame .................. 50
2.8 SecondPriceAuction................... 51
2.9 TheLandlordandtheEvictionNotice ......... 51
2.10 Hagar’s Battles . ..................... 52
2.11AnIncreasingBidAuction................ 52
2.12FootballStrategy ..................... 52
2.13 A Military Strategy Game . . .............. 52
2.14StrategicVoting...................... 53
2.15TheDebtorandHisCreditors.............. 54
2.16 Eliminating Dominated Strategies ad Absurdum .... 54
2.17PokerwithBlung.................... 55
2.18TheCentipedeGame................... 56
3 Playing it Straight: Pure Strategy Nash Equilibria 58
3.1 Introduction........................ 58
3.2 PureCoordinationGames ................ 59
3.3 CompetitiononMainStreet............... 60
3.4 APureCoordinationGame ............... 60
3.5 TwinSisters........................ 60
3.6 VariationsonDuopoly .................. 61
3.7 TheTobaccoMarket ................... 62
3.8 TheWeWillNotBeUndersoldGame ......... 62
3.9 TheKlingonsandtheSnarks .............. 63
3.10Chess|TheTrivialPastime ............... 64
3.11TheSamaritan’sDilemma ................ 64
3.12TheRottenKidTheorem ................ 65
3.13TheIllogicofConﬂictEscalation ............ 66
3.14HowtoValueLotteries.................. 67
3.14.1Where’sJack? .................... 68
3.15 The Expected Utility Principle . . . . . . . ...... 68
3.16BuyingFireInsurance .................. 69
3.17NeoclassicalEconomicsandGameTheory ....... 70
3.18 Markets as Disciplining Devices: Allied Widgets . . . . 73
3.19TheTruthGame ..................... 78
3.20TheShopperandtheFishMerchant .......... 79
3.21CommonKnowledge ................... 80
3.22TheWomenofSevitan.................. 80
4 Catching ’em O Guard: Mixed Strategy Nash Equilib-
ria 82
4.1 Introduction........................ 82
4.2 MixedStrategies:BasicDenitions ........... 83
4.3 The Fundamental Theorem of Mixed Strategy Nash
Equilibrium ........................ 84
4.4 SolvingforMixedStrategyNashEquilibria ...... 85
4.5 ReviewingtheTerminology ............... 86
4.6 BigMonkeyandLittleMonkeyRevisited........ 87
4.7 DominanceRevisted ................... 87
4.8 CompetitiononMainStreetRevisited ......... 87
4.9 BattleoftheSexes .................... 87
4.10ThrowingFingers..................... 88
4.11One-card,Two-roundPokerwithBlung ....... 88
4.11.1 SimulatingOne-cardTwo-roundBlu ....... 89
4.12BehavioralStrategiesinExtensiveFormGames.... 90
4.13LionsandAntelope.................... 92
4.14 The Santa F eBar..................... 93
4.15 Orange-Throat, Blue-Throat and Yellow-Striped Lizards 94
4.16SexRatiosasNashEquilibria .............. 95
4.17TennisStrategy...................... 96
4.18AMatingGame...................... 97
4.19HardLove......................... 98
4.20CoordinationFailureI .................. 98
4.21AdvertisingGame..................... 98
4.22PreservationofEcologyGame.............. 99
4.23ColonelBlottoGame................... 99
4.24NumberGuessingGame .................100
4.25TargetSelection......................100
4.26AReconnaissanceGame .................101
4.27AttackonHiddenObject.................101
4.28Two-PersonZero-SumGames ..............102
4.29 An Introduction to Forward Induction . . . . . ....103
4.30 Mutual Monitoring and Cooperation in a Partnership . 104
4.31MutualMonitoringandCooperationinTeams.....105
4.32Altruism(?)inBirdFlocks................106
4.33RobinHoodandLittleJohn...............107
4.34TheMotorist’sDilemma .................107
4.35FamilyPolitics ......................108
4.36FrankieandJohnny....................108
4.37ACardGame.......................108
4.38Cheater-Inspector.....................109
4.39TheGrouchoMarxGame ................109
4.40RealMenDon’tEatQuiche ...............110
4.41TheVindicationoftheHawk ..............111
4.42CorrelatedEquilibria...................112
4.43PokerwithBlungRevisited ..............114
4.44 The Equivalence of Behavioral and Mixed Strategies . 114
5 Moving Through the Game Tree: Subgame Perfection,
Incredible Threats, and Trembling Hands 117
5.1 Introduction........................117
5.2 SubgamePerfection....................119
5.3 StackelbergLeadership..................122
5.4 TheSubwayEntryDeterrenceGame ..........123
5.5 TheDr.StrangeloveGame................123
5.6 The St ahl-RubinsteinBargainingModel ........124
5.7 Huey,Dewey,andLouisSplitaDollar .........126
5.8 TheLittleMissMuetGame ..............126
5.9 NuisanceSuits.......................127
5.10 Cooperation in an Overlapping Generations Economy . 129
5.11TheFinitelyRepeatedPrisoner’sDilemma.......130
5.12 The Finitely Repeated Prisoner’s Dilemma: the Fine
Points ...........................136
5.13FuzzySubgamePerfection................137
5.14 Perfect Behavioral Nash Equilibria and Bayesian Up-
dating ...........................139
5.15Selten’sHorse.......................141
5.16TremblingHandPerfection................142
5.17 Nature Abhors Low Probability Events . . . ......143
6 Repeated Games, Trigger Strategies, and Tacit Collu-
sion 145
6.1 Introduction........................145
6.2 BigFishandLittleFish .................146
6.3 TacitCollusion ......................148
6.4 The Folk Theorem: An Embarr asdeRichesses.....153
6.5 VariationsontheFolkTheorem.............154
6.6 TheOneStageDeviationPrinciple ...........156
6.7 A Trembling Hand, Repeated Game, Cooperative
Equilibrium ........................156
6.8 Death and Discount Rates in Repeated Games . . . . 158
6.9 MannyandMoe......................158
6.10TheStrategyofanOilCartel ..............158
6.11Tit-for-Tat.........................159
6.12APublicGoodsExperiment...............160
6.13ReputationalEquilibrium ................161
6.14ContingentRenewalContracts..............161
6.14.1 Costless Contract Enforcement: Achilles’ Heel of
NeoclassicalEconomics ...............163
6.14.2 Contingent Renewal Markets Do Not Clear in
Equilibrium......................163
6.14.3MoneyConfersShort-sidePower ..........166
6.14.4WhenMoneyTalks,PeopleListen .........166
6.14.5 The Economy is Controlled by the Wealthy . . . . 167
6.15ContingentRenewalLaborMarkets...........167
6.16I’dRatherSwitchthanFight ..............171
7 Biology Meets Economics: Evolutionary Stability and
the Origins of Dynamic Game Theory 175
7.1 The Origins of Evolutionary Stability . . . . . . ....175
7.2 PropertiesofEvolutionaryStableStrategies ......176
7.3 When are Evolutionarily Stable Strategies Unbeatable? 179
7.4 CooperativeFishing ...................179
7.5 A Nash Equilibrium that is not Evolutionarily Stable . 180
7.6 Rock, Paper, and Scissors is not Evolutionarily Stable 180
7.7 SexRatiosasEvolutionaryStableStrategies......181
7.8 InvasionofthePureStrategyMutants .........181
7.9 MultipleEvolutionaryStableStrategies ........181
7.10TheLogicofAnimalConﬂict ..............182
7.11Hawks,DovesandBourgeois...............185
7.12TrogsandFarfel......................185
7.13 Evolutionary Stability in Finite Populations . . ....187
7.14 Evolutionary Stability in Asymmetric Games . ....188

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2010-3-18 15:41:33

8 Dynamical Systems and Dierential Equations 191
8.1 Introduction........................191
8.2 DynamicalSystems....................192
8.3 PopulationGrowth ....................193
8.4 Population Growth with Limited Carrying Capacity . 194
8.5 TheLotka-VolterraPredator-PreyModel........195
8.6 DynamicalSystemsTheory ...............199
8.6.1 ExistenceandUniqueness..............200
8.6.2 TheLinearizationTheorem.............201
8.7 DynamicalSystemsinOneDimension .........202
8.8 DynamicalSystemsinTwoDimensions.........204
8.9 ExercisesinTwo-DimensionalLinearSystems.....208
8.10CulturalDynamics ....................209
8.11 A Lotka-Volterra Model with Limited Carrying Capacity210
8.12TakeNoPrisoners ....................210
8.13TheHartman-GrobmanTheorem............211
8.14 Special Features of Two-Dimensional Dynamical Sys-
tems ............................212
8.15 Non-Hyperbolic Dynamical Systems in Two Dimensions213
8.16 Liapunov’s Theorem . . . . . ..............213

9 Evolutionary Dynamics 215
9.1 Introduction........................215
9.2 TheOriginsofEvolutionaryDynamics .........216
9.2.1 StrategiesasReplicators...............217
9.2.2 ADynamicHawk/DoveGame ...........220
9.2.3 Sexual Reproduction, Biological Fitness, and the
ReplicatorDynamic .................222
9.3 PropertiesoftheReplicatorEquation..........224
9.4 Characterizing the Two-Variable Replicator Dynamic . 225
9.5 Do Dominated Strategies Survive under a Replicator
Dynamic? .........................226
9.6 Nash Equilibria and Stability under a Replicator Dy-
namic ...........................227
9.7 Evolutionary Stability and Evolutionary Equilibrium . 229
9.8 Bayesian Perfection, Stable Sets and the Replicator
Dynamic..........................230
9.9 InvasionofthePureStrategyMutants,II .......230
9.10 A Generalization of Rock, Paper, and Scissors . . . . . 231
9.11 Uta Stansburia in Motion ................232
9.12 The Dynamics of Rock-Paper-Scissors and Related
Games ...........................233
9.13 Replicator Dynamics, the Lotka-Volterra Model, and
Biodiversity ........................234
9.14AsymmetricEvolutionaryGames ............236
9.15 Asymmetric Evolutionary Games: Reviewing the Troups240
9.16TheEvolutionofTrustandHonesty ..........240
9.17TheLoraxesandThoraxes................242
9.18 The Replicator Dynamic, Cultural Transmission, and
SocialImitation......................243
10 Markov Economies and Stochastic Dynamical Systems 246
10.1Introduction........................246
10.2 The Emergence of a Money in a Markov Economy . . 247
10.2.1 SimulatingAMonetaryEconomy..........251
10.3GoodVibrations .....................254
10.4AdaptiveLearning ....................256
10.4.1TheSteadyStateofaMarkovChain........258
10.5 Adaptive Learning When not all Conventions are Equal259
10.6 Adaptive Learning in a Pure Coordination Game with
Errors ...........................260

10.7 Stochastic Stability . . . . . . ..............261
11 Homo Reciprocans, Homo Egualis and other Contribu-
tors to the Human Behavioral Repertoire 263
11.1Introduction........................263
11.2ModelingtheHumanActor ...............265
11.2.1 Interpreting the Results of Experimental Game
Theory ........................267
11.2.2 Self-InterestandRationality.............269
11.3 Behavioral Economics: Games against Nature and
AgainstOurselves.....................270
11.3.1 Time Inconsistency and Hyperbolic Discounting . 271
11.3.2 Choice Under Uncertainty: Logic and Heuristics . 273
11.3.3 Loss Aversion and Status Quo Bias.........276
11.4 Experimental Game Theory: The Laboratory Meets
StrategicInteraction ...................277
11.4.1TheUltimatumGame................279
11.4.2ThePublicGoodsGame...............280
11.4.3ThePublicGoodsGamewithRetaliation.....282
11.4.4TheCommonPoolResourceGame.........282
11.5HomoEgualis .......................284
11.6 Homo Reciprocans: Modeling Strong Reciprocity . . . 287
11.7AltruismandAssortativeInteractions .........292
11.8TheEvolutionofStrongReciprocity ..........297
11.9 Homo Parochius: Modeling Insider-Outsider Relations 303
12 Learning Who Your Friends Are: Bayesian Games and
Private Information 309
12.1PrivateInformationandBayesianGames........309
12.2 The Role of Beliefs in Games with Private Information 313
12.3HagglingattheBazaar..................316
12.4AdverseSelection.....................319
12.5AMarketforLemons...................320
12.6ChoosinganExorcist...................321
12.7AFirstPriceSealed-BidAuction ............324
12.8ACommonValueAuction:TheWinner’sCurse....325
12.9CommonValueAuctionII................325
12.10Predatory Pricing: Pooling and Separating Equilibria . 327
12.11LimitPricing .......................329
12.12ASimpleLimitPricingModel..............331

13 When it Pays to be Truthful: Signaling Games and Com-
munications Equilibria 332
13.1 Signalling as a Coevolutionary Process . . . ......332
13.2 A Generic Signalling Game . . . . . . . . . ......333
13.3IntroductoryOers....................335
13.4WebSites(forSpiders) .................335
13.5 Sex and Piety: The Darwin-Fisher Model of Runaway
SexualSelection......................337
13.6BiologicalSignalsasHandicaps .............342
13.7TheShepherdswhoNeverCryWolf ..........344
13.8MyBrother’sKeeper...................346
13.9HonestSignallyamongPartialAltruists ........348
13.10Educational Signalling I . ................351
13.11EducationasAScreeningDevice ............353
13.12Capital as a Signaling Device: A Model of Productiv-
ityEnhancingRedistributions..............355
14 Bosses and Workers, Landlords and Peasants, and other
Principal-Agent Models 357
14.1IntroductiontothePrincipal-AgentModel.......357
14.2LaborDisciplinewithMonitoring............358
14.3LaborasGiftExchange .................360
14.4LaborDisciplinewithProtSignaling .........361
14.4.1 Properties of the Labor Discipline Model . . . . . 364
14.5PeasantsandLandlords .................365
14.6Mr.Smith’sCarInsurance................366
14.7AGenericOne-ShotPrincipalAgentModel ......367
15 Axiomatic and Game Theoretic Models of Bargaining 370
15.1Introduction........................370
15.2TheNashBargainingModel...............371
15.3 Risk Aversion and the Nash Bargaining Solution . . . 374
15.4 The St ahl-Rubinstein Bargaining Model with Respo-
nentOutsideOptions...................376
15.5BargainingwithTwo-SidedOutsideOptions......377
15.6 How the St ahl-Rubinstein Bargaining Model Becomes
theNashBargainingModel ...............378
15.7 Zeuthen Lotteries and the Nash Bargaining Solution . 379
15.8BargainingwithFixedCosts...............380
15.9BargainingwithIncompleteInformation ........381

16 Probability and Decision Theory 382
16.1 Probability Spaces . . . . . . ..............382
16.2 Conditional Probability . . . . ..............383
16.3Bayes’Rule ........................383
16.4DrugTesting .......................384
16.5ABoltFactory ......................384
16.6 Color Blindness . .....................384
16.7Urns ............................384
16.8TheMontyHallGame ..................384
16.9 The Logic of Murder and Abuse . . . . . . . . . ....386
16.9.1ThePrincipleofInsucientReason ........388
16.10Ah,ThoseKids ......................388
16.11TheGreensandtheBlacks................388
16.12Laplace’sLawofSuccession ...............389
16.13TheBrainandKidneyProblem.............389
16.14SexualHarassmentontheJob..............389
16.15TheValueofEye-WitnessTestimony..........389
16.16TheEndoftheWorld ..................390
16.17BillandHarry.......................390
16.18WhenWeaknessisStrength ...............391
16.19MarkovChains ......................391
16.19.1TheErgodicTheoremforMarkovChains .....397
16.19.2TheSisyphusianMarkovChain...........398
16.19.3Andrei Andreyevich’s Two Urn Problem . . ....399
16.20Preferences and Expected Utility . . . . . . . . ....400
16.21Exceptions to the Expected Utility Principle . . ....405
16.22Risk Behavior and the Shape of the Utility Function . 406