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Insights Into Game Theory
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Contents Preface page xi Introduction xii 1 Mathematical Matching 1 1.1 Introduction 1 1.2 The Matching Problem 2 1.3 Exercises 7 1.4 Further Examples 10 1.5 Exercises 13 1.6 A Procedure for Finding Stable Matching Systems (The Gale–Shapley Algorithm) 15 1.7 Exercises 18 1.8 A Stable Matching System Always Exists 19 1.9 The Maximum Number of Courtship Stages in the Gale–Shapley Algorithm 21 1.10 Generalization 26 1.11 Exercises 33 1.12 The Gale–Shapley Algorithm and the Assignment Problem 37 1.13 Exercises 40 1.14 Optimality 43 1.15 Exercises 49 1.16 Condition for the Existence of a Unique Stable Matching System 52 1.17 Exercises 54 1.18 Discussion 55 1.19 Review Exercises 56 viii contents 2 Social Justice 59 2.1 Presentation of the Problem 59 2.2 Mathematical Description of the Problem 62 2.3 Exercises 64 2.4 Social Choice Function 67 2.5 Axioms for the Social Choice Function 77 2.6 Exercises 80 2.7 What Follows from Axioms 1–4? 81 2.8 Exercises 85 2.9 Arrow’s Theorem 87 2.10 What Next? 92 2.11 Review Exercises 93 3 The Shapley Value in Cooperative Games 97 3.1 Introduction 97 3.2 Cooperative Games 98 3.3 Important Examples of Coalition Function Games 101 3.4 Exercises 105 3.5 Additive Games 106 3.6 Superadditive Games 107 3.7 Majority Games 108 3.8 Exercises 112 3.9 Symmetric Players 113 3.10 Exercises 115 3.11 Null Players 116 3.12 Exercises 117 3.13 The Sum of Games 118 3.14 Exercises 121 3.15 The Shapley Value 124 3.16 Exercises 133 3.17 Dissolving a Partnership 133 3.18 Exercises 141 contents ix 3.19 The Shapley Value as the Average of Players’ Marginal Contributions 142 3.20 Exercises 146 3.21 The Shapley Value as a Player’s Index of Power in Weighted Majority Games 148 3.22 Exercises 153 3.23 The Shapley–Shubik Index as an Index for the Analysis of Parliamentary Phenomena 153 3.24 Exercises 156 3.25 The Security Council 156 3.26 Exercises 158 3.27 Cost Games 159 3.28 Exercises 162 3.29 Review Exercises 164 4 Analysis of a Bankruptcy Problem from the Talmud 166 4.1 Introduction 166 4.2 The Contested Garment 168 4.3 Exercises 171 4.4 A Physical Interpretation of the Contested-Garment Principle 172 4.5 Exercises 176 4.6 A Bankruptcy Problem from the Talmud 177 4.7 Exercises 180 4.8 Existence and Uniqueness 182 4.9 Divisions Consistent with the Contested-Garment Principle 186 4.10 Exercises 191 4.11 Consistency 192 4.12 Exercises 194 4.13 Rif’s Law of Division 194 4.14 Exercises 196 4.15 Proportional Division 196 x contents 4.16 O’Neill’s Law of Division 197 4.17 Exercises 200 4.18 Discussion 201 4.19 Review Exercises 203 A Answers to the Exercises 205 A.1 Chapter 1 205 A.2 Chapter 2 213 A.3 Chapter 3 220 A.4 Chapter 4 229 Bibliography 233 Index 235 |
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