Table Of ContentTheory and Decision Library C 45
Game Theory, Social Choice, Decision Theory, and Optimization
Holger Ingmar Meinhardt
The Pre-Kernel as
a Tractable Solution
for Cooperative
Games
An Exercise in Algorithmic Game Theory
THEORYAND DECISIONLIBRARY C
GameTheory,Social Choice, DecisionTheory,and Optimization
Volume45
Editors-in-Chief
HervéMoulin,Glasgow,Scotland,UnitedKingdom
HansPeters,Maastricht,TheNetherlands
HonoraryEditor
StefH.Tijs,Tilburg,TheNetherlands
EditorialBoard
Jean-JacquesHerings,Maastricht,TheNetherlands
MatthewO.Jackson,Stanford,CA,USA
MamuroKaneko,Tokyo,Japan
HansKeiding,Copenhagen,Denmark
BezalelPeleg,Jerusalem,Israel
ClemensPuppe,Karlsruhe,Germany
AlvinE.Roth,Stanford,CA,USA
DavidSchmeidler,TelAviv,Israel
ReinhardSelten,Bonn,Germany
WilliamThomson,Rochester,NJ,USA
RakeshVohra,Evanston,IL,USA
PeterWakker,Rotterdam,TheNetherlands
Forfurthervolumes:
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Holger Ingmar Meinhardt
The Pre-Kernel
as a Tractable Solution
for Cooperative Games
An Exercise in Algorithmic Game Theory
123
HolgerIngmarMeinhardt
InstituteofOperationsResearch
KarlsruheInstituteofTechnology(KIT)
Karlsruhe
Germany
ISSN0924-6126 ISSN2194-3044(electronic)
ISBN978-3-642-39548-2 ISBN978-3-642-39549-9(eBook)
DOI10.1007/978-3-642-39549-9
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013950138
(cid:2)c Springer-VerlagBerlinHeidelberg2014
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Foreword
This monograph is devoted to the pre-kernel scheme as an attractive and fair
solution concept which can be used alternatively to the Shapley value in order
to divide the outcomes of mutual cooperation. The framework of the class of
cooperativegames without transferable utility is used to analyze a process of fair
division in accordance with the pre-kernel. Here convex analysis plays a central
role,in particulartheoperationofFenchelconjugates.Pursuinga betterandmore
comprehensiveunderstandingofafaircompromise,asetofprinciplesofdistributive
justice is presented. On the one hand the axiomatic foundation of the Shapley
value is given that lets one consider this solution scheme as a fair division rule.
On the other handthe axiomaticfoundationof the pre-kernelis presentedto set a
counterpointto that what subjects consider as fair or unfair. Besides its attractive
axiomatic characterization,it is arguedthat an efficientand easy computabilityof
a proposed solution is a desirable feature to qualify as a fair division rule. It is
illustratedonthebasisofanexamplefromcooperativeoligopolygamethatthepre-
kernelistechnicallymorecomplicatedandisthereforedifficulttocompute,aclear
disadvantagewith regardtothe Shapleyvaluewhichpossessesa simple,efficient,
andsystematiccomputationprocedure.
Adualrepresentationmethod,basedonFenchelconjugation,givesrisetoafull
characterizationofthepre-kernelintermsofconstrainedminimumsets,whichcan
betreatedbyprojectionmethods.Thisapproachisbasedonthepioneeringworkof
Martínez-Legaz(1996).Inthiscontextitisprovedthatunderaregimeoforthogonal
projections,thesequenceofpayoffvectorsgeneratedbyeachoneoftheproposed
computationmethodsiscycle-free.
Inordertoevaluatetheperformanceoftheproposedalgorithms,someempirical
resultsarediscussedandaresetincomparisontosomewell-knowncomputational
methodsfromtheliterature.
Most parts of the monograph are new and yet unpublished. The book is an
undeniablereferenceforabetterunderstandingoftherelationshipbetweenthepre-
kernelschemeandtheShapleyvalueconcept.
Karlsruhe,Germany DiethardPallaschke
March2013
v
Preface
This book provides an alternative approach to study the pre-kernel solution of
transferable utility games based on a generalized conjugation theory from con-
vex analysis. Although the pre-kernel solution possesses an appealing axiomatic
foundation that lets one consider this solution concept as a standard of fairness
withoutreferringto thenotionof“interpersonalcomparisonofutility” –ithas, in
addition,afoundationthatisbasedonanoncooperativebargainingmodel–many
scholars,nevertheless,regardthepre-kernelanditsrelatedsolutionsasobscureand
too technically complex to be treated as a real alternative to the Shapley value.
Comprehensibleandefficientcomputabilityiswidelyregardedasadesirablefeature
to qualify a solution concept apart from its axiomatic foundationas a standard of
fairness.Subjectswilldistrustadivisionrulethatcanonlybecomputedand“fully
understood”by a handfulof experts. It remains the flavor of being fleeced by the
proposerormediator.Incasethatdecisionmakershavethefeelingofbeingtreated
unfairly,theywillobstructanagreement.Especiallyiftheyarenotabletocompute
theirownsharebythemselves.Here,theShapleyvalue–ifitisacceptedasafair
divisionrule–offerstheclearadvantagethateverybodycancalculatehisownshare
simplybyapplyingitsformula.
A central task in real-life managerial problems concerns the sustainability of
the resource. Especially in areas where multilateral agreements are nonbinding,
compliance becomes a crucial issue in order to avoid the destruction of a natural
resource. Compliance can be achieved when agents exercise self-constraint and
refrain from using their powers to exploit one another. Then a solution can
be obtained that is acceptable for all participants: a fair compromise. Such a
compromise will be considered a fair outcome when it produces a common
virtual world where compliance is reality and obstruction is held to account.
Rather than considering fairness as some opaque concept, in order to advance
ourunderstandingofcomplianceon nonbindingagreements,we studyfairnesson
the set of principles (axioms) which describe the pre-kernel. However, the pre-
kernel is not fully understood to give a simple and efficient computation method
to qualify as an attractive fair division rule. Other rules of distributive justice are
usually considered more attractive for implementing fairness in economical or in
vii
viii Preface
experimentalsituations,whichmight,however,haveoriginatedinitssimplicityof
computationandnotonitsnormativelycompellingfoundation.
In thismonograph,we reviewandthenimproveanapproachthatwasinvented
byMeseguer-Artola(1997)tocomputethepre-kernelofacooperativegamebythe
indirectfunction.TheindirectfunctionisknownastheFenchel-Moreauconjugation
of the characteristic function introducedby Martínez-Legaz(1996).By following
andextendingtheindirectfunctionapproachproposedbyMeseguer-Artola(1997),
we are able to characterize the pre-kernel of the grand coalition simply by the
solution sets of a family of quadratic objective functions.Now, each functioncan
be solved by means well known from analysis and linear algebra. For economic
situationswhichofferstrongincentiveformutualcooperation,theformerlydifficult
issue of a fair divisionof the proceedsin accordancewith the pre-kernelcan now
be easily accomplished and verified by partners involved in a common venture.
Henceforth, the rules of distributive justice (axioms) which characterize the pre-
kernelasafaircompromiseshallbedeemedasanalternativetosolveandstabilize
theself-administrationproblemofanaturalresource.Formanysubjectsfairnessisa
nebulousnotion.Subjectshavedifferentperceptionsofwhattheyconsidertobefair
orunfair.Enlargingthesetofrulesfordistributivejusticethatcanbedevotedand
agreeduponforself-managinganaturalresourcemighthelptoavoiditsdestruction.
Karlsruhe,Germany HolgerIngmarMeinhardt
March2013
Acknowledgements
Iamgratefultoanumberofpeoplewhoofferedvaluablesuggestionsandremarks
during the preparation of the monograph.Especially, I am very much indebted to
Diethard Pallaschke for his careful mathematical proofreading and the advice he
offeredduringdifferentstages of the monograph.I am also verythankfulto Axel
Ostmann and Jean Derks for their helpful comments and constructive proposals
of improvement to an earlier version of the manuscript. For their stimulating
discussion, I want to express my gratitude to Chih Chang, Theo Driessen, Walter
Kern, and Martha Saboyá Baquero. Furthermore, I want to thank my brother and
sister-in-law, Bjoern and Lynnette Meinhardt, for their linguistic assistance. Of
course,fortheerrorsthatremaininthiswork,theauthoracceptsfullresponsibility.
In addition,I acknowledgethe kind and excellentsupportfrom the staff of the
Steinbuch Center of Computation (SCC) at the Karlsruhe Institute of Technology
(KIT)theyofferedduringthesoftwaredevelopmentanditsimplementation.Special
thanks go pars pro toto to Horst Gernert, Hartmut Häfner, Elisabeth Syrjakow,
RichardWalter,andPaulWeber.
Thisbookiswarmlydedicatedtomyparents.
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