The Cooperative Game Theory of Networks and Hierarchies (eBook)

Robert P. Gilles (Autor)

eBook Download: PDF

2010
XI, 270 Seiten
Springer Berlin (Verlag)
978-3-642-05282-8 (ISBN)

Infos

The book brings together an overview of standard concepts in cooperative game theory with applications to the analysis of social networks and hierarchical authority organizations. The standard concepts covered include the multi-linear extension, the Core, the Shapley value, and the cooperative potential. Also discussed are the Core for a restricted collection of formable coalitions, various Core covers, the Myerson value, value-based potentials, and share potentials. Within the context of social networks this book discusses the measurement of centrality and power as well as allocation rules such as the Myerson value and hierarchical allocation rules. For hierarchical organizations, two basic approaches to the exercise of authority are explored; for each approach the allocation of the generated output is developed. Each chapter is accompanied by a problem section, allowing this book to be used as a textbook for an advanced graduate course on game theory.

Inhalt

Preface 6
Contents 10
List of Figures 12
to 1 Cooperative Game Theory 13
1.1 Aims and Outline of the Book 14
1.2 Game Forms 17
1.2.1 The Normal Form 19
1.2.2 The Characteristic Function Form 21
1.3 Cooperative Games 24
1.3.1 Basic Properties of Cooperative Games 25
1.3.2 The Standard Basis 26
1.3.3 The Unanimity Basis 27
1.3.4 Essential Games and Imputations 30
1.4 Multilinear Extensions 32
1.5 Problems 37
to 2 The Core of a Cooperative Game 40
2.1 Basic Properties of the Core 41
2.1.1 Representing the Core of a Three Player Game 42
2.1.2 The Core and Domination 46
2.1.3 Existence of Core Imputations 48
2.2 The Core Based on a Collection of Coalitions 50
2.2.1 Balanced Collections 56
2.2.2 Strongly Balanced Collections 58
2.2.3 Lattices and Hierarchies 59
2.3 Core Covers and Convex Games 62
2.3.1 The Weber Set 62
2.3.2 The Selectope 64
2.4 Appendix: Proofs of the Main Theorems 66
2.5 Problems 79
to 3 Axiomatic Value Theory 82
3.1 Equivalent Formulations of the Shapley Value 83
3.2 Three Axiomatizations of the Shapley Value 87
3.2.1 Shapley's Axiomatization 88
3.2.2 Young's Axiomatization 91
3.2.3 van den Brink's Axiomatization 93
3.3 The Shapley Value as a Utility Function 94
3.4 The Myerson Value 100
3.5 Appendix: Proofs of the Main Theorems 103
3.6 Problems 112
to 4 The Cooperative Potential 118
4.1 A Potential Function for Cooperative Games 119
4.2 The Cooperative Potential and the Shapley Value 123
4.3 Consistency and the Reductionist Approach 126
4.4 Beyond the Cooperative Potential 130
4.4.1 Value-Based Potentials 130
4.4.2 Share Functions and Share Potentials 133
4.5 Appendix: Proofs of the Main Theorems 137
4.6 Problems 149
to 5 Directed Communication Networks 154
5.1 Directed Networks 156
5.2 Measuring Dominance in Directed Networks 158
5.2.1 The Degree Measure 159
5.2.2 The ?-Measure 162
5.2.3 Iterated Power Measures 169
5.3 Hierarchical Allocation Rules on Network Games 174
5.3.1 Cooperative Network Situations 175
5.3.2 Network Myerson Values 177
5.3.3 The Hierarchical Payoff Property 179
5.3.4 The -Hierarchical Value 182
5.4 Appendix: Proofs of the Main Theorems 186
5.5 Problems 197
to 6 Cooperative Theories of Hierarchical Organizations 199
6.1 Games with a Permission Structure 201
6.1.1 The Conjunctive Approach 204
6.1.2 The Disjunctive Approach 208
6.2 Shapley Permission Values 220
6.2.1 The Conjunctive Permission Value 220
6.2.2 The Disjunctive Permission Value 226
6.3 Modeling Economic Phenomena 229
6.4 Appendix: Proofs of the Main Theorems 232
6.5 Discussion: More About the Disjunctive Restriction 247
6.6 Problems 249
References 254
Index 260

Produkt-Details

• Erscheint lt. Verlag: 2.4.2010
• Reihe/Serie: Theory and Decision Library C
• Reihe/Serie: Theory and Decision Library C
• Zusatzinfo: XI, 270 p. 20 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Authority Structures • cooperative game theory • Game Theory • Hierarchies • Organizations • Social Networks
• ISBN-10: 3-642-05282-7 / 3642052827
• ISBN-13: 978-3-642-05282-8 / 9783642052828