ebook img

Partition Function Form Games. Coalitional Games with Externalities PDF

317 Pages·2018·2.42 MB·english
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Partition Function Form Games. Coalitional Games with Externalities

László Á. Kóczy Partition Function Form Games Coalitional Games with Externalities LászlóÁ.Kóczy InstituteofEconomics CentreforEconomicsandRegionalStudies HungarianAcademyofSciences Budapest Hungary and KeletiFacultyofBusinessandManagement ÓbudaUniversity Budapest Hungary ISSN0924-6126 Theory and DecisionLibraryC ISBN978-3-319-69840-3 ISBN978-3-319-69841-0(eBook) https://doi.org/10.1007/978-3-319-69841-0 LibraryofCongressControlNumber:2018933585 ©SpringerInternationalPublishingAG2018 Preface I became interested in partition function form games early in my academic career and, by now, have a few publications on the topic. Such games are not new but it wasoftendifficulttopresentthesepapersatconferencesastheaudienceknewlittle about partition function form games. I often wished I could direct the interested researchertoabasictextforaquickintroduction.ItwasLarsEhlerswhosuggested thatperhapsIshouldwriteabookaboutthem.Theoriginalplanwaslessambitious but,goingdeeperintheliterature,manynewconnections,hiddenresults,andpapers havebeenuncovered.Itwasalsodifficulttokeepupwiththelatestadditionstothe fieldwithsomanynewpaperspublishedinthelastfewyears.Completingthisbook tookseveralyearsanditwouldnothavebeenpossiblewithoutthepatienceofmy employer, the help of my colleagues and my family. I am grateful to the Institute of Economics, Centre for Economic and Regional Sciences, Hungarian Academy of Sciences and O´buda University for letting me devote time to this project and to my game theory friends and colleagues for tolerating my anti-social behaviour regardingseminarsandrefereeingrequests.IespeciallythankHansPeters,Miklo´s Pinte´randA´gnesCsehfortheircommentsandAttilaNa´sforcarefullyreadingthe manuscript.Last,butnotleast,Ithankmyfamily:mywife,AnitaVizyandchildren Lacko´,Mici,Palko´ andIstifortheirloveandendlesspatience. Budapest,Hungary La´szlo´ A´.Ko´czy June2017 Contents ListofSymbols .................................................... xiii PartI Foundations 1 Purposeofthebook ............................................ 3 1.1 Introduction ............................................... 3 1.2 AboutthisBook ........................................... 5 References..................................................... 6 2 TerminologyandNotation ...................................... 7 2.1 Introduction ............................................... 7 2.2 Numbers.................................................. 8 2.3 Sets ...................................................... 9 2.3.1 Partitions ........................................... 9 2.3.2 Permutations........................................ 14 2.4 VectorsandMatrices........................................ 15 2.5 GraphsandNetworks ....................................... 16 2.5.1 Connectivity ........................................ 16 2.5.2 Trees .............................................. 17 References..................................................... 17 3 Preliminaries .................................................. 19 3.1 Introduction ............................................... 19 3.2 NormalFormGames........................................ 20 3.3 ExtensiveFormGames...................................... 22 3.4 CharacteristicFunctionFormGames .......................... 25 3.4.1 Properties .......................................... 27 3.4.2 SolutionConcepts:Stability ........................... 29 3.4.3 SolutionConcepts:Fairness ........................... 37 3.4.4 GamesoverNetworks ................................ 41 3.5 PartitionFunctionFormGames............................... 43 3.5.1 ExternalitiesinPartitionFunctionFormGames........... 45 3.5.2 PropertiesofPartitionFunctionFormGames............. 47 3.5.3 ModifiedPartitionFunctionFormGames ................ 50 3.6 DynamicCooperativeGames................................. 53 References..................................................... 55 PartII Stability 4 Dominance .................................................... 63 4.1 Introduction ............................................... 63 4.2 DirectDominanceModels ................................... 64 4.3 CoalitionFormationRules ................................... 66 4.4 Farsightedness ............................................. 67 References..................................................... 68 5 TheCore...................................................... 71 5.1 Introduction ............................................... 71 5.2 Thea-Core ............................................... 72 5.3 Thew-Core ............................................... 74 5.4 PartialAgreementEquilibriumandthes-Core .................. 76 5.4.1 PartialAgreementEquilibriumandtheg-Core............ 77 5.4.2 Thes-Core.......................................... 78 5.4.3 TheDisintegrationRule............................... 80 5.4.4 Thef-Core ......................................... 80 5.5 Themergeorm-Core ....................................... 80 5.6 TheProjectionRuleandthed-core............................ 81 5.7 TheRationalExpectationsCore .............................. 82 5.8 RecursiveCores............................................ 83 5.8.1 Ther-Theory........................................ 84 5.8.2 TheRecursiveCore .................................. 87 5.9 AnAxiomaticComparison................................... 99 5.9.1 Axioms ............................................101 5.9.2 Characterisations ....................................102 5.10 Conclusion................................................103 References.....................................................104 6 ImplementationoftheCore .....................................107 6.1 Introduction ...............................................107 6.2 SimultaneousAnnouncementModels..........................109 6.3 AlternatingOffersBargainingModel ..........................110 6.4 SequentialCoalitionFormationwithFixedPayoffDivision .......112 6.4.1 StationaryEquilibriumCoalitionStructures ..............112 6.4.2 SymmetricPartitionFunctionFormGames ..............114 6.5 SequentialCoalitionFormationinTotallyBalancedGames .......116 6.5.1 AContinuous-TimeBargainingModel ..................117 6.5.2 AnOrder-IndependentEquilibriumCoalitionStructures....119 6.6 ImplementationinGeneralPartitionFunctionFormGames .......122 6.6.1 ATwo-StageCoalitionFormationGame.................123 6.6.2 ASequentialCoalitionFormationModel ................125 6.7 Conclusion................................................137 6.A Appendix .................................................138 6.A.1 TheCoreoftheGameinExample6.21..................138 References.....................................................140 7 Otherstabilityconcepts.........................................143 7.1 Introduction ...............................................143 7.2 TheStableSet .............................................143 7.2.1 Introduction.........................................143 7.3 EquilibriumBindingAgreements .............................145 7.3.1 StrategicSetting .....................................146 7.3.2 PartitionFunctionSetting .............................148 7.3.3 Extensions..........................................148 7.4 Conclusion................................................149 References.....................................................149 PartIII Fairness 8 Axioms .......................................................153 8.1 Introduction ...............................................153 8.2 CommonAxioms ..........................................154 8.2.1 Efficiency ..........................................154 8.2.2 Anonymity .........................................155 8.2.3 Symmetry ..........................................156 8.2.4 StandardProperty....................................157 8.3 CarriersandNullPlayers ....................................157 8.3.1 CarrierAxiom.......................................157 8.3.2 Oligarchy...........................................158 8.3.3 NullPlayers.........................................159 8.3.4 DummyPlayers .....................................163 8.4 Additivity,LinearityandTransfer .............................164 8.5 MarginalityandMonotonicity................................165 8.6 ExternalAxioms ...........................................167 8.7 Conclusion................................................170 References.....................................................170 9 TheShapley-value..............................................173 9.1 Introduction ...............................................173 9.2 ClassicalAdditivity-BasedAxiomatisations ....................174 9.2.1 Anonymity,CarrierandAdditivity......................174 9.3 NullPlayerPropertyVariations...............................175 9.3.1 NullPlayerProperty..................................176 9.3.2 StrongNullPlayerProperty ...........................176 9.3.3 Steady-MarginalityNullPlayerProperty.................177 9.3.4 WeakNullPlayerPropertyandMarginality ..............179 9.4 Marginality-BasedAxiomatisations ...........................181 9.5 AveragingMethods.........................................182 9.5.1 TheAverageApproach ...............................182 9.5.2 NullPlayera ........................................183 9.6 AxiomatisationwithoutMarginalityorNullPlayerProperty.......185 9.7 GeneralisationtoGamesthatareNotFullyCohesive.............186 9.8 Potential ..................................................186 9.8.1 MarginalityandPotential .............................187 9.8.2 ExtendingPotentialtoPartitionFunctionFormGames.....187 9.8.3 RestrictionOperators.................................189 9.8.4 Consistency.........................................192 9.9 Implementation:AnEfficientMulti-BiddingProcedure...........194 9.10 Calculations ...............................................197 9.11 Conclusion................................................198 References.....................................................199 10 Othervalues...................................................201 10.1 Introduction ...............................................201 10.2 TheBanzhaf-value .........................................202 10.2.1 SimpleGames.......................................202 10.2.2 PowerMeasures .....................................203 10.2.3 Extensions..........................................204 10.2.4 TheAverageApproach ...............................209 10.2.5 AnOrdinalApproach.................................211 10.3 TheConsensusValue .......................................212 10.3.1 Definition ..........................................212 10.3.2 Characterisation .....................................213 10.4 CoalitionFormationValues ..................................214 10.4.1 Model..............................................214 10.4.2 Values .............................................215 10.5 ExternalitiesinNetworks....................................216 10.5.1 AGeneralisedShapley-value ..........................216 10.5.2 ThePositionValue ...................................218 10.6 Conclusion................................................219 References.....................................................220 PartIV Applications 11 OligopolyModels ..............................................225 11.1 Introduction ...............................................225 11.2 CoalitionFormationinCournotOligopolies ....................226 11.2.1 WhydoFirmsMerge?................................226 11.2.2 MultipleMergers ....................................227 11.2.3 TheFormalModel ...................................228 11.3 SymmetricCournotMergerSystem ...........................231 11.3.1 ADiscretePartitionFunctionModel ....................231 11.3.2 TheStabilityoftheMonopoly .........................232 11.4 MergerswithAntitrustRegulation ............................234 11.5 MergersinInternationalOligopolies...........................235 References.....................................................236 12 EnvironmentalApplications.....................................239 12.1 Introduction ...............................................239 12.2 InternationalEnvironmentalAgreements.......................240 12.3 TragedyoftheCommonsinaFishingLake.....................244 12.3.1 Theory .............................................244 12.3.2 Simulations .........................................247 12.4 StraddlingStockFisheries ...................................249 12.4.1 Bioeconomicmodel ..................................250 12.4.2 Partitionfunction ....................................250 12.5 SharingaRiver ............................................253 12.5.1 ThePractice ........................................254 12.5.2 AFormalModel.....................................255 12.5.3 DownstreamIncrementalDistribution ...................256 12.5.4 UpstreamIncrementalDistribution .....................258 12.5.5 TheWeightedHierarchicalDistribution .................258 12.5.6 SatiableAgents......................................261 12.5.7 Conclusion .........................................265 References.....................................................266 13 FemtocellNetworks ............................................269 13.1 Introduction ...............................................269 13.2 JointFrequencyScheduling..................................270 13.2.1 TheModel..........................................271 13.2.2 Simulations .........................................274 13.2.3 InterferencewithMacrocellUserEquipment .............275 13.3 InterferenceAlignment......................................276 13.3.1 TheModel..........................................276 13.3.2 Simulations .........................................278 References.....................................................278 14 FurtherApplications ...........................................281 14.1 Introducion................................................281 14.2 GamesoverUtilityNetworks.................................282 14.2.1 TrafficRouting ......................................282 14.2.2 ElectricalPowerTransmissionNetworks ................283 14.3 MatchingwithExternalities..................................284 14.4 FreeTradeAgreementsandCustomsUnions ...................285 References.....................................................286 References.........................................................289 Index .............................................................303 Symbols complexconjugate,page8 (cid:1) differencebetweenapartitionandaset,page14 n differencebetweenpartitions,page14 n setdifference,page9 n elementofasetrelation,page9 22S powersetofS,page9 accessibility,page54 ! sequentialdomination,page97 ! inductionrelationbetweenaprocessandascenario,page215 ! strictsubsetrelation,page9 (cid:26) subsetrelation,page9 (cid:18) finestcommoncoarsening,page12 _ maximumoperator,page165 _ coarsestcommonrefinement,page12 ^ minimumoperator,page165 ^ absolutevalue,page8 j(cid:1)j partiton partitionsize,page10 jh ij set setcardinality,page9 jh ij [ set ] partitionintosingletons,page10 h i upperintegerpart,page8 d(cid:1)e partition minimalpartitionsize,page10 bh ic integerpart,page8 b(cid:1)c () positivepart,page8 (cid:1) + domd d-domination,page66 dom dominancerelation,page31 domd d-dominationviaM,page66 M dom dominancerelationviaS,page31 S = equality,page15 > vectorrelation:strictlygreater,coordinate-wisegreaterthanequal, page15 vectorrelation:greaterthanorequal,page15 (cid:21)

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.