Insights into Game Theory: an Alternative Mathematical Experience

Cover......Page 1
Half-title......Page 2
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Introduction......Page 14
1.1 Introduction......Page 17
1.2 The matching problem......Page 18
1.3 Exercises......Page 23
1.4 Further examples......Page 26
1.5 Exercises......Page 29
The Gale–Shapley algorithm for finding a stable matching system......Page 31
1.7 Exercises......Page 34
1.8 A stable matching system always exists......Page 35
1.9 The maximum number of courtship stages in the gale–shapley algorithm......Page 37
I. The number of men does not equal the number of women......Page 42
II. Existence of a preference list that does not include all members of the opposite sex......Page 44
III. Possible Indifference......Page 45
1.11 Exercises......Page 49
The medical school admissions problem......Page 53
1.13 Exercises......Page 56
1.14 Optimality......Page 59
1.15 Exercises......Page 65
1.16 Condition for the existence of a unique stable matching system......Page 68
1.17 Exercises......Page 70
1.18 Discussion......Page 71
1.19 Review exercises......Page 72
2.1 Presentation of the problem......Page 75
2.2 Mathematical description of the problem......Page 78
2.3 Exercises......Page 80
2.4 Social choice function......Page 83
I. Majority Rule......Page 84
III. Decision by a Dictator......Page 86
IV. Ill-defined “Rule”......Page 87
V. “Just” Rule......Page 88
VI. Dependence on Irrelevant Alternatives......Page 90
VII. Positive Association of Individual Preferences and Social Preferences......Page 92
Axiom 1: The Domain and Range of the Function......Page 93
Axiom 2: Positive Association of Individual Preferences and the Social Preference......Page 94
Axiom 5: Non-dictatorship......Page 95
2.6 Exercises......Page 96
2.7 What follows from axioms 1–4?......Page 97
2.8 Exercises......Page 101
2.9 Arrow\'s theorem......Page 103
2.10 What next?......Page 108
2.11 Review exercises......Page 109
3.1 Introduction......Page 113
3.2 Cooperative games......Page 114
3.3 Important examples of coalition function games......Page 117
3.4 Exercises......Page 121
3.5 Additive games......Page 122
3.6 Superadditive games......Page 123
3.7 Majority games......Page 124
3.8 Exercises......Page 128
3.9 Symmetric players......Page 129
3.10 Exercises......Page 131
3.11 Null players......Page 132
3.12 Exercises......Page 133
3.13 The sum of games......Page 134
3.14 Exercises......Page 137
3.15 The shapley value......Page 140
3.17 Dissolving a partnership......Page 149
3.18 Exercises......Page 157
3.19 The shapley value as the average of players\' marginal contributions......Page 158
3.20 Exercises......Page 162
3.21 The shapley value as a player\'s index of power in weighted majority games......Page 164
3.23 The shapley–shubik index as an index for the analysis of parliamentary phenomena......Page 169
3.25 The security council......Page 172
3.26 Exercises......Page 174
3.27 Cost games......Page 175
3.28 Exercises......Page 178
3.29 Review exercises......Page 180
4.1 Introduction......Page 182
4.2 The contested garment......Page 184
Mathematical generalization:......Page 186
4.3 Exercises......Page 187
4.4 A physical interpretation of the contested-garment principle......Page 188
4.5 Exercises......Page 192
4.6 A bankruptcy problem from the talmud......Page 193
4.7 Exercises......Page 196
4.8 Existence and uniqueness......Page 198
4.9 Divisions consistent with the contested-garment principle......Page 202
4.10 Exercises......Page 207
4.11 Consistency......Page 208
4.13 Rif\'s law of division......Page 210
4.15 Proportional division......Page 212
4.16 O\'neill\'s law of division......Page 213
4.17 Exercises......Page 216
4.18 Discussion......Page 217
4.19 Review exercises......Page 219
A.1 Chapter 1......Page 221
A.2 Chapter 2......Page 229
A.3 Chapter 3......Page 236
A.4 Chapter 4......Page 245
Bibliography......Page 249
Index......Page 251