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RepeatedGames Threeleadingexpertshaveproducedalandmarkworkbasedonasetofworking paperspublishedbytheCenterforOperationsResearchandEconometrics(CORE) at Universite´ Catholique de Louvain in 1994, under the title “Repeated Games,” which holds almost mythic status among game theorists. Jean-François Mertens, SylvainSorin,andShmuelZamirhavesignificantlyelevatedtheclarityanddepth of presentation with many results presented at a level of generality that goes far beyondtheoriginalpapers–manywrittenbytheauthorsthemselves.Numerous resultsarenew,andmanyclassicresultsandexamplesarenottobefoundelsewhere. Mostremainstateoftheartintheliterature.Thisbookisfullofchallengingand important problems that are set up as exercises, with detailed hints provided for theirsolution.Anewbibliographytracesthedevelopmentofthecoreconceptsup tothepresentday. The late Jean-François Mertens (11 March 1946–17 July 2012) was professor at the Universite´ Catholique de Louvain (where he earned his PhD) and a member of the CORE. One of the world’s leading experts in game theory and economic theory,Mertensistheauthorofseminalpapersonequilibriumselectioningames, formulation of Bayesian analysis, repeated and stochastic games, general equilibrium, social choice theory, and dynamic general equilibrium. A Fellow of the Econometric Society, he was also a founding member of the Center for Game TheoryinEconomicsattheStateUniversityofNewYorkatStonyBrook. Sylvain Sorin is a member of the Mathematics Department at the Universite´ PierreetMarieCurie.HewaspreviouslyprofessoratUniversite´L.Pasteur,Stras- bourg,andUniversite´ ParisX-Nanterre.Hehasbeenanaffiliatedmemberofthe De´partementdeMathe´matiques,EćoleNormaleSupe´rieure(Paris);theLaboratoire d’Econome´trie,EćolePolytechnique,Palaiseau;andtheCenterforGameTheory inEconomics,StateUniversityofNewYorkatStonyBrook.HeisaFellowofthe EconometricSocietyandwasachartermemberoftheGameTheorySocietyand editor-in-chiefoftheInternationalJournalofGameTheory. Shmuel Zamir, author of the textbook Game Theory (with M. Maschler and E. Solan, Cambridge 2013), is professor emeritus at the Hebrew University of JerusalemandafoundingmemberoftheCenterfortheStudyofRationalitythere. He is professor of economics at the University of Exeter Business School, UK. ZamirisaFellowoftheEconometricSociety,achartermemberandaformercoun- cilmemberoftheGameTheorySociety,andanaffiliatedmemberoftheCenterfor GameTheoryinEconomicsattheStateUniversityofNewYorkatStonyBrook. Since2008,hehasbeentheeditor-in-chiefoftheInternationalJournalofGame Theory. EconometricSocietyMonographs Editors: RosaL.Matzkin,UniversityofCalifornia,LosAngeles GeorgeJ.Mailath,UniversityofPennsylvania TheEconometricSocietyisaninternationalsocietyfortheadvancementofeconomictheoryin relationto statistics and mathematics. The EconometricSociety Monographseries is designed to promote the publication of original research contributions of high quality in mathematical economicsandtheoreticalandappliedeconometrics. OtherTitlesintheSeries: G.S.Maddala,Limiteddependentandqualitativevariablesineconometrics,9780521241434, 9780521338257 GerardDebreu,Mathematicaleconomics:TwentypapersofGerardDebreu,9780521237369, 9780521335614 Jean-MichelGrandmont,Moneyandvalue:Areconsiderationofclassicalandneoclassical monetaryeconomics,9780521251419,9780521313643 FranklinM.Fisher,Disequilibriumfoundationsofequilibriumeconomics,9780521378567 AndreuMas-Colell,Thetheoryofgeneralequilibrium:Adifferentiableapproach, 9780521265140,9780521388702 TrumanF.Bewley,Editor,Advancesineconometrics–FifthWorldCongress(VolumeI), 9780521467261 TrumanF.Bewley,Editor,Advancesineconometrics–FifthWorldCongress(VolumeII), 9780521467254 Herve´Moulin,Axiomsofcooperativedecisionmaking,9780521360555,9780521424585 L.G.Godfrey,Misspecificationtestsineconometrics:TheLagrangemultiplierprincipleand otherapproaches,9780521424592 TonyLancaster,Theeconometricanalysisoftransitiondata,9780521437899 AlvinE.RothandMarildaA.OlivieraSotomayor,Editors,Two-sidedmatching:Astudyin game-theoreticmodelingandanalysis,9780521437882 WolfgangHa¨rdle,Appliednonparametricregression,9780521429504 Jean-JacquesLaffont,Editor,Advancesineconomictheory–SixthWorldCongress(VolumeI), 9780521484596 Jean-JacquesLaffont,Editor,Advancesineconomictheory–SixthWorldCongress(VolumeII), 9780521484602 HalbertWhite,Estimation,inferenceandspecification,9780521252805,9780521574464 ChristopherSims,Editor,Advancesineconometrics–SixthWorldCongress(VolumeI), 9780521444590,9780521566100 ChristopherSims,Editor,Advancesineconometrics–SixthWorldCongress(VolumeII), 9780521444606,9780521566094 RogerGuesnerie,Acontributiontothepuretheoryoftaxation,9780521629560 DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeI),9780521589833 DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeII),9780521589826 DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeIII),9780521580137,9780521589819 DonaldP.Jacobs,EhudKalai,andMortonI.Kamien,Editors,Frontiersofresearchineconomic theory:TheNancyL.SchwartzMemorialLectures,1983–1997,9780521632225, 9780521635387 A.ColinCameronandPravinK.Trivedi,Regressionanalysisofcountdata,9780521632010, 9780521635677 SteinarStrom,Editor,Econometricsandeconomictheoryinthe20thcentury:TheRagnarFrisch CentennialSymposium,9780521633239,9780521633659 Continuedfollowingtheindex Repeated Games Jean-François Mertens Sylvain Sorin Universite´PierreetMarieCurie,France Shmuel Zamir TheHebrewUniversityofJerusalem,Israel,andtheUniversityof Exeter,UK 32AvenueoftheAmericas,NewYork,NY10013-2473,USA CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107662636 ©Jean-FrançoisMertens,SylvainSorin,andShmuelZamir2015 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2015 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Mertens,Jean-François,1946– Repeatedgames/Jean-FrançoisMertens,SylvainSorin,ShmuelZamir. pages cm.–(Econometricsocietymonographs;55) Includesbibliographicalreferencesandindex. ISBN978-1-107-03020-6(hardback)–ISBN978-1-107-66263-6(paperback) 1.Gametheory. 2.Stochasticprocesses. I.Sorin,Sylvain. II.Zamir,Shmuel. III.Title. HB144.M47 2014 519.3–dc23 2014006953 ISBN978-1-107-03020-6Hardback ISBN978-1-107-66263-6Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguarantee thatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate. Contents ListofFigures pagexiii ForewordbyRobertJ.Aumann xv Preface xxi Acknowledgments xxiii PresentationoftheContent xxv PartA:BackgroundMaterial I BasicResultsonNormalFormGames 3 I.1 TheMinmaxTheorem 4 I.1.a DefinitionsandNotations 4 I.1.b ABasicTheorem 5 I.1.c Convexity 6 I.1.d MixedStrategies 6 I.1.e NoteontheSeparationTheorem 9 Exercises 10 I.2 ComplementstotheMinmaxTheorem 15 I.2.a TheTopologyonS 16 I.2.b LackofContinuity:Regularization 16 I.2.c LackofCompactness:Approximation 19 I.2.d Measurability:SymmetricCase 20 I.2.e PureOptimalStrategies 23 Exercises 24 I.3 TheMinmaxTheoremforOrderedFields 31 Exercises 32 I.4 EquilibriumPoints 45 Exercises 45 vii viii Contents II BasicResultsonExtensiveFormGames 58 II.1 TheExtensiveForm 58 II.1.a Definitions 58 II.1.b TheFiniteCase 60 II.1.c AMeasurableSetup 62 Exercises 64 II.2 InfiniteGames 90 II.2.a InfiniteGameswithPerfectInformation 90 II.2.b Remarks:InfiniteGameswithoutPerfect Information 95 Exercises 96 II.3 CorrelatedEquilibriaandExtensions 101 II.3.a CorrelatedEquilibria 101 II.3.b MultistageGames,ExtensiveFormCorrelated Equilibria 103 II.3.c CommunicationEquilibria 105 II.3.d FiniteGames 108 Exercises 110 II.4 VectorPayoffs 118 Exercises 121 III TheBeliefSpace 123 III.1 TheUniversalBeliefSpace 123 III.1.a StatesoftheWorldandTypes 124 III.1.b BeliefSubspaces 135 III.2 ConsistencyandCommonKnowledge 138 III.3 AnApproximationTheorem 151 III.4 GameswithIncompleteInformation 153 III.4.a TheModel 153 III.4.b Two-PersonZero-SumCase 154 III.4.c “Approachability”inOne-ShotGames 158 III.4.d ConcavificationandConvexification 161 Exercises 166 IV GeneralModelofRepeatedGames 171 IV.1 TheModel 171 IV.1.a States,Signals,andTransitions 172 IV.1.b StrategiesandPayoffs 173 IV.1.c Zero-SumCase 174 IV.1.d Non-Zero-SumCase 176 IV.1.e StochasticGamesandGameswithIncomplete Information 177 Exercises 178