Partition Function Form Games. Coalitional Games with Externalities

317 Pages·2018·2.42 MB·english

by  László Á. Kóczy

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)