Michael Maschler z"l was one of the best teachers that I have ever had (I took his "differential equations" course in Spring 1972).
He was also a legend among high school students, for his engaging outreach lectures that he gave from time to time, and indeed was very interested
in all levels of mathematics education. When I came across this introduction to Game Theory, I had very high expectations.
I am glad to say that this book even exceeded them! Maschler must have transmitted his exceptional pedagogical talents to his disciple, Ein-Ya Gura.
Together they produced a masterpiece of pedagogy that would appeal to a very wide range of potential readers, from the bright high school student
all the way to the non-expert professional mathematician. Even game-theorists would find a new perspective to familiar material, and this book would help them
to better present this material to their students. This book can also serve as a paradigm for future text-book writing, and shows how one can
be rigorous without being pedantic, engaging without being frivolous, and the many beautifully chosen exercises will guarantee that students will
actually learn to do mathematics, and slowly, but surely, digest very sophisticated material that earned Nobel prizes to their discoverers.
Author(s): Ein-Ya Gura, Michael Maschler
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 249
Cover......Page 1
Half-title......Page 0
Title......Page 3
Copyright......Page 4
Dedication......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 11
1.1 Introduction......Page 14
1.2 The matching problem......Page 15
1.3 Exercises......Page 20
1.4 Further examples......Page 23
1.5 Exercises......Page 26
The Gale–Shapley algorithm for finding a stable matching system......Page 28
1.7 Exercises......Page 31
1.8 A stable matching system always exists......Page 32
1.9 The maximum number of courtship stages in the gale–shapley algorithm......Page 34
I. The number of men does not equal the number of women......Page 39
II. Existence of a preference list that does not include all members of the opposite sex......Page 41
III. Possible Indifference......Page 42
1.11 Exercises......Page 46
The medical school admissions problem......Page 50
1.13 Exercises......Page 53
1.14 Optimality......Page 56
1.15 Exercises......Page 62
1.16 Condition for the existence of a unique stable matching system......Page 65
1.17 Exercises......Page 67
1.18 Discussion......Page 68
1.19 Review exercises......Page 69
2.1 Presentation of the problem......Page 72
2.2 Mathematical description of the problem......Page 75
2.3 Exercises......Page 77
2.4 Social choice function......Page 80
I. Majority Rule......Page 81
III. Decision by a Dictator......Page 83
IV. Ill-defined “Rule”......Page 84
V. “Just” Rule......Page 85
VI. Dependence on Irrelevant Alternatives......Page 87
VII. Positive Association of Individual Preferences and Social Preferences......Page 89
Axiom 1: The Domain and Range of the Function......Page 90
Axiom 2: Positive Association of Individual Preferences and the Social Preference......Page 91
Axiom 5: Non-dictatorship......Page 92
2.6 Exercises......Page 93
2.7 What follows from axioms 1–4?......Page 94
2.8 Exercises......Page 98
2.9 Arrow's theorem......Page 100
2.10 What next?......Page 105
2.11 Review exercises......Page 106
3.1 Introduction......Page 110
3.2 Cooperative games......Page 111
3.3 Important examples of coalition function games......Page 114
3.4 Exercises......Page 118
3.5 Additive games......Page 119
3.6 Superadditive games......Page 120
3.7 Majority games......Page 121
3.8 Exercises......Page 125
3.9 Symmetric players......Page 126
3.10 Exercises......Page 128
3.11 Null players......Page 129
3.12 Exercises......Page 130
3.13 The sum of games......Page 131
3.14 Exercises......Page 134
3.15 The shapley value......Page 137
3.17 Dissolving a partnership......Page 146
3.18 Exercises......Page 154
3.19 The shapley value as the average of players' marginal contributions......Page 155
3.20 Exercises......Page 159
3.21 The shapley value as a player's index of power in weighted majority games......Page 161
3.23 The shapley–shubik index as an index for the analysis of parliamentary phenomena......Page 166
3.25 The security council......Page 169
3.26 Exercises......Page 171
3.27 Cost games......Page 172
3.28 Exercises......Page 175
3.29 Review exercises......Page 177
4.1 Introduction......Page 179
4.2 The contested garment......Page 181
Mathematical generalization:......Page 183
4.3 Exercises......Page 184
4.4 A physical interpretation of the contested-garment principle......Page 185
4.5 Exercises......Page 189
4.6 A bankruptcy problem from the talmud......Page 190
4.7 Exercises......Page 193
4.8 Existence and uniqueness......Page 195
4.9 Divisions consistent with the contested-garment principle......Page 199
4.10 Exercises......Page 204
4.11 Consistency......Page 205
4.13 Rif's law of division......Page 207
4.15 Proportional division......Page 209
4.16 O'neill's law of division......Page 210
4.17 Exercises......Page 213
4.18 Discussion......Page 214
4.19 Review exercises......Page 216
A.1 Chapter 1......Page 218
A.2 Chapter 2......Page 226
A.3 Chapter 3......Page 233
A.4 Chapter 4......Page 242
Bibliography......Page 246
Index......Page 248
