Insights into Game Theory: An Alternative Mathematical Experience

252 Pages·2008·3.28 MB·English

by  Ein-Ya Gura

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Insights into Game Theory
Fewbranchesofmathematicshavebeenmoreinfluentialinthe
socialsciencesthangametheory.Inrecentyears,ithasbecomean
essentialtoolforallsocialscientistsstudyingthestrategicbehavior
ofcompetingindividuals,firms,andcountries.However,the
mathematicalcomplexityofgametheoryisoftenveryintimidating
forstudentswhohaveonlyabasicunderstandingofmathematics.
InsightsintoGameTheoryaddressesthisproblembyproviding
studentswithanunderstandingofthekeyconceptsandideasof
gametheorywithoutusingformalmathematicalnotation.The
authorsusefourverydifferenttopics(collegeadmissions,social
justiceandmajorityvoting,coalitionsandcooperativegames,anda
bankruptcyproblemfromtheTalmud)toinvestigatefourareasof
gametheory.Theresultisafascinatingintroductiontotheworldof
gametheoryanditsincreasinglyimportantroleinthesocial
sciences.
ein-yaguraisaSeniorLecturerattheRothbergInternational
SchoolandamemberoftheCenterfortheStudyofRationalityat
theHebrewUniversityofJerusalem.
michaelb.maschlerisaProfessorofMathematicsandamember
oftheCenterfortheStudyofRationalityattheHebrewUniversity
ofJerusalem.ProfessorMaschlerisaworld-renownedgametheorist.
HehashadalongandfruitfulcollaborationwithRobertAumann
(co-winnerofthe2005NobelPrizeforEconomics),includingtheir
solutiontoa2000-year-oldpuzzleoninheritancelawsinthe
Talmud,discussedinthisbook.

Insights into Game Theory
An Alternative Mathematical
Experience
ein-ya gura
and
michael maschler
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521874229
©Ein-Ya Gura and Michael B. Maschler 2008
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-45538-4 eBook (EBL)
ISBN-13 978-0-521-87422-9 hardback
ISBN-13 978-0-521-69692-0 paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
ThisbookisdedicatedtothememoryofMichaelMaschler,
whopassedawayonJuly20,2008.

Contents
Preface page xi
Introduction xii
1 MathematicalMatching 1
1.1 Introduction 1
1.2 TheMatchingProblem 2
1.3 Exercises 7
1.4 FurtherExamples 10
1.5 Exercises 13
1.6 AProcedureforFindingStableMatchingSystems
(TheGale–ShapleyAlgorithm) 15
1.7 Exercises 18
1.8 AStableMatchingSystemAlwaysExists 19
1.9 TheMaximumNumberofCourtshipStagesinthe
Gale–ShapleyAlgorithm 21
1.10 Generalization 26
1.11 Exercises 33
1.12 TheGale–ShapleyAlgorithmandthe
AssignmentProblem 37
1.13 Exercises 40
1.14 Optimality 43
1.15 Exercises 49
1.16 ConditionfortheExistenceofaUniqueStable
MatchingSystem 52
1.17 Exercises 54
1.18 Discussion 55
1.19 ReviewExercises 56
viii contents
2 SocialJustice 59
2.1 PresentationoftheProblem 59
2.2 MathematicalDescriptionoftheProblem 62
2.3 Exercises 64
2.4 SocialChoiceFunction 67
2.5 AxiomsfortheSocialChoiceFunction 77
2.6 Exercises 80
2.7 WhatFollowsfromAxioms1–4? 81
2.8 Exercises 85
2.9 Arrow’sTheorem 87
2.10 WhatNext? 92
2.11 ReviewExercises 93
3 TheShapleyValueinCooperativeGames 97
3.1 Introduction 97
3.2 CooperativeGames 98
3.3 ImportantExamplesofCoalitionFunctionGames 101
3.4 Exercises 105
3.5 AdditiveGames 106
3.6 SuperadditiveGames 107
3.7 MajorityGames 108
3.8 Exercises 112
3.9 SymmetricPlayers 113
3.10 Exercises 115
3.11 NullPlayers 116
3.12 Exercises 117
3.13 TheSumofGames 118
3.14 Exercises 121
3.15 TheShapleyValue 124
3.16 Exercises 133
3.17 DissolvingaPartnership 133
3.18 Exercises 141