Springer07《合作博弈导论》（Introduction to the Theory of Cooperative Games 2nd ed）Introduction to the Theory of Cooperative Games (Theory and Decision Library C) (Hardcover)

by Bezalel Peleg (Author), Peter Sudhölter (Author)

 

Hardcover: 328 pages

Publisher: Springer; 2nd ed. edition (October 3, 2007)

Language: English

Book Description

 

This book systematically presents the main solutions of cooperative games: the core, bargaining set, kernel, nucleolus, and the Shapley value of TU games, and the core, the Shapley value, and the ordinal bargaining set of NTU games. To each solution the authors devote a separate chapter wherein they study its properties in full detail. Moreover, important variants are defined or even intensively analyzed. The authors also investigate in separate chapters continuity, dynamics, and geometric properties of solutions of TU games. The study culminates in uniform and coherent axiomatizations of all the foregoing solutions (excluding the bargaining set). Such axiomatizations have not appeared in any book. Moreover, the book contains a detailed analysis of the main results on cooperative games without side payments. Such analysis is very limited or non-existent in other books on game theory.

 

 

　　　　 

 

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

Notation and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVII

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Cooperative Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 TU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 A Guide for the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Special Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Interpersonal Comparisons of Utility . . . . . . . . . . . . . . . . 5

1.3.3 Nash’s Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Part I TU Games

2 Coalitional TU Games and Solutions . . . . . . . . . . . . . . . . . . . . . 9

2.1 Coalitional Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Some Families of Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Market Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Cost Allocation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 The Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 The Bondareva-Shapley Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 An Application to Market Games . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Totally Balanced Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Some Families of Totally Balanced Games . . . . . . . . . . . . . . . . . 35

3.4.1 Minimum Cost Spanning Tree Games . . . . . . . . . . . . . . . 35

3.4.2 Permutation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 A Characterization of Convex Games . . . . . . . . . . . . . . . . . . . . . 39

3.6 An Axiomatization of the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 An Axiomatization of the Core on Market Games . . . . . . . . . . 42

3.8 The Core for Games with Various Coalition Structures . . . . . . 44

3.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 The Bargaining Set M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Existence of the Bargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Balanced Superadditive Games and the Bargaining Set . . . . . . 62

4.4 Further Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.1 The Reactive and the Semi-reactive Bargaining Set . . . 65

4.4.2 The Mas-Colell Bargaining Set . . . . . . . . . . . . . . . . . . . . . 69

4.5 Non-monotonicity of Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . 72

4.6 The Bargaining Set and Syndication: An Example . . . . . . . . . . 76

4.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 The Prekernel, Kernel, and Nucleolus . . . . . . . . . . . . . . . . . . . . 81

5.1 The Nucleolus and the Prenucleolus . . . . . . . . . . . . . . . . . . . . . . 82

5.2 The Reduced Game Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Desirability, Equal Treatment, and the Prekernel . . . . . . . . . . . 89

5.4 An Axiomatization of the Prekernel . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Individual Rationality and the Kernel . . . . . . . . . . . . . . . . . . . . . 94

5.6 Reasonableness of the Prekernel and the Kernel . . . . . . . . . . . . 98

5.7 The Prekernel of a Convex Game . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.8 The Prekernel and Syndication . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 The Prenucleolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.1 A Combinatorial Characterization of the Prenucleolus . . . . . . . 108

6.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 An Axiomatization of the Prenucleolus . . . . . . . . . . . . . . . . . . . . 112

6.3.1 An Axiomatization of the Nucleolus . . . . . . . . . . . . . . . . 115

6.3.2 The Positive Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 The Prenucleolus of Games with Coalition Structures . . . . . . . 119

6.5 The Nucleolus of Strong Weighted Majority Games . . . . . . . . . 120

6.6 The Modiclus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6.1 Constant-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.6.2 Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.6.3 Weighted Majority Games . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Geometric Properties of the ε-Core, Kernel, and Prekernel 133

7.1 Geometric Properties of the ε-Core . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Some Properties of the Least-Core . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 The Reasonable Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4 Geometric Characterizations of the Prekernel and Kernel . . . . 142

7.5 A Method for Computing the Prenucleolus . . . . . . . . . . . . . . . . 146

7.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

X Contents

8 The Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.1 Existence and Uniqueness of the Value . . . . . . . . . . . . . . . . . . . . 152

8.2 Monotonicity Properties of Solutions and the Value . . . . . . . . . 156

8.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.4 The Potential of the Shapley Value . . . . . . . . . . . . . . . . . . . . . . . 161

8.5 A Reduced Game for the Shapley Value . . . . . . . . . . . . . . . . . . . 163

8.6 The Shapley Value for Simple Games . . . . . . . . . . . . . . . . . . . . . 168

8.7 Games with Coalition Structures . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.8 Games with A Priori Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.9 Multilinear Extensions of Games . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.11 A Summary of Some Properties of the Main Solutions . . . . . . . 179

9 Continuity Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 181

9.1 Upper Hemicontinuity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 181

9.2 Lower Hemicontinuity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 184

9.3 Continuity of the Prenucleolus . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

10 Dynamic Bargaining Procedures for the Kernel and the

Bargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10.1 Dynamic Systems for the Kernel and the Bargaining Set . . . . . 190

10.2 Stable Sets of the Kernel and the Bargaining Set . . . . . . . . . . . 195

10.3 Asymptotic Stability of the Nucleolus . . . . . . . . . . . . . . . . . . . . . 198

10.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Part II NTU Games

11 Cooperative Games in Strategic and Coalitional Form . . . . 203

11.1 Cooperative Games in Strategic Form . . . . . . . . . . . . . . . . . . . . . 203

11.2 α- and β-Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

11.3 Coalitional Games with Nontransferable Utility . . . . . . . . . . . . 209

Contents XI

11.4 Cooperative Games with Side Payments but Without

Transferable Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

11.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

12 The Core of NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

12.1 Individual Rationality, Pareto Optimality, and the Core . . . . . 214

12.2 Balanced NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

12.3 Ordinal and Cardinal Convex Games. . . . . . . . . . . . . . . . . . . . . . 220

12.3.1 Ordinal Convex Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

12.3.2 Cardinal Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

12.4 An Axiomatization of the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

12.4.1 Reduced Games of NTU Games . . . . . . . . . . . . . . . . . . . . 224

12.4.2 Axioms for the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

12.4.3 Proof of Theorem 12.4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

12.5 Additional Properties and Characterizations . . . . . . . . . . . . . . . 230

12.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

13 The Shapley NTU Value and the Harsanyi Solution . . . . . . 235

13.1 The Shapley Value of NTU Games. . . . . . . . . . . . . . . . . . . . . . . . 235

13.2 A Characterization of the Shapley NTU Value . . . . . . . . . . . . . 239

13.3 The Harsanyi Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

13.4 A Characterization of the Harsanyi Solution . . . . . . . . . . . . . . . 247

13.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

14 The Consistent Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

14.1 For Hyperplane Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

14.2 For p-Smooth Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

14.3 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

14.3.1 The Role of IIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

14.3.2 Logical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

14.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

XII Contents

15 On the Classical Bargaining Set and the Mas-Colell

Bargaining Set for NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . 269

15.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

15.1.1 The Bargaining Set M. . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

15.1.2 The Mas-Colell Bargaining Set
     MB and Majority

Voting Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

15.1.3 The 3 × 3 Voting Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 274

15.2 Voting Games with an Empty Mas-Colell Bargaining Set . . . . 278

15.3 Non-levelled NTU Games with an Empty Mas-Colell

Prebargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

15.3.1 The Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

15.3.2 Non-levelled Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

15.4 Existence Results for Many Voters . . . . . . . . . . . . . . . . . . . . . . . . 289

15.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

16 Variants of the Davis-Maschler Bargaining Set for NTU

Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

16.1 The Ordinal Bargaining Set
     Mo . . . . . . . . . . . . . . . . . . . . . . . . . . 295

16.2 A Proof of Billera’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

16.3 Solutions Related to Mo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

16.3.1 The Ordinal Reactive and the Ordinal Semi-Reactive

Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

16.3.2 Solutions Related to the Prekernel . . . . . . . . . . . . . . . . . . 303

16.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

 

 

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