Introduction to the theory of cooperative games / by Bezalel Peleg and Peter Sudhölter.

Format:

Book

Author/Creator:

Peleg, Bezalel.

Contributor:

Sudhölter, Peter.

Series:

Theory and decision library. Game theory, mathematical programming, and operations research ; Series C, v. 34.

Theory and decision library. Series C, Game theory, mathematical programming, and operations research ; v. 34

Language:

English

Subjects (All):

Game theory.

Group decision making.

Cooperation.

Physical Description:

xvi, 378 pages : illustrations ; 25 cm.

Place of Publication:

Boston : Kluwer Academic Publishers, [2003]

Summary:

Introduction to the Theory of Cooperative Games systematically studies 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 a separate chapter is devoted, in which its properties are investigated in full detail. Moreover, important variants are defined or even intensively analyzed. Separate chapters cover continuity, dynamics, and geometric properties of solutions of TU games. This study culminates in uniform and coherent axiomatizations of all the foregoing solutions (excluding the bargaining set). Except for the Shapley value such axiomatizations have not appeared in any book. Moreover, Introduction to the Theory of Cooperative Games contains a detailed analysis of the main results on cooperative games without side payments. Such analysis is very limited or non-existent in the existing literature on game theory.

Contents:

1.1 Cooperative Games 1

1.2.1 TU Games 3

1.2.2 NTU Games 5

1.3.1 Axiomatizations 6

1.3.2 Interpersonal Comparisons of Utility 7

1.3.3 Nash's Program 7

Part I TU Games

2 Coalitional TU Games and Solutions 11

2.1 Coalitional Games 12

2.2 Some Families of Games 16

2.2.1 Market Games 16

2.2.2 Cost Allocation Games 18

2.2.3 Simple Games 21

2.3 Properties of Solutions 24

3 The Core 35

3.1 The Bondareva-Shapley Theorem 36

3.2 An Application to Market Games 42

3.3 Totally Balanced Games 44

3.4 Some Families of Totally Balanced Games 46

3.4.1 Minimum Cost Spanning Tree Games 46

3.4.2 Permutation Games 47

3.5 A Characterization of Convex Games 50

3.6 An Axiomatization of the Core 52

3.7 An Axiomatization of the Core on Market Games 55

3.8 The Core for Games with Various Coalition Structures 58

4 Bargaining Sets 67

4.1 The Bargaining Set M 68

4.2 Existence of the Bargaining Set 75

4.3 Balanced Superadditive Games and the Bargaining Set 82

4.4 Further Bargaining Sets 85

4.4.1 The Reactive and the Semi-Reactive Bargaining Set 85

4.4.2 The Mas-Colell Bargaining Set 91

4.5 Non-Monotonicity of Bargaining Sets 96

4.6 The Bargaining Set and Syndication An Example 101

5 The Prekernel, Kernel, and Nucleolus 107

5.1 The Nucleolus and the Prenucleolus 108

5.2 The Reduced Game Property 114

5.3 Desirability, Equal Treatment, and the Prekernel 119

5.4 An Axiomatization of the Prekernel 121

5.5 Individual Rationality and the Kernel 125

5.6 Reasonableness of the Prekernel and the Kernel 131

5.7 The Prekernel of a Convex Game 133

5.8 The Prekernel and Syndication 137

6 The Prenucleolus 143

6.1 A Combinatorial Characterization of the Prenucleolus 144

6.2 Preliminary Results 146

6.3 An Axiomatization of the Prenucleolus 149

6.3.1 An Axiomatization of the Nucleolus 154

6.3.2 The Positive Core 156

6.4 The Prenucleolus of Games with Coalition Structures 159

6.5 The Nucleolus of Strong Weighted Majority Games 160

6.6 The Modiclus 166

6.6.1 Constant-Sum Games 172

6.6.2 Convex Games 173

6.6.3 Weighted Majority Games 174

7 Geometric Properties of the [varepsilon]-Core, Kernel, and Prekernel 177

7.1 Geometric Properties of the [varepsilon]-Core 178

7.2 Some Properties of the Least-Core 181

7.3 The Reasonable Set 184

7.4 Geometric Characterizations of the Prekernel and Kernel 189

7.5 A Method for Computing the Prenucleolus 194

8 The Shapley Value 201

8.1 Existence and Uniqueness of the Value 202

8.2 Monotonicity Properties of Solutions and the Value 208

8.3 Consistency 212

8.4 The Potential of the Shapley Value 215

8.5 A Reduced Game for the Shapley Value 218

8.6 The Shapley Value for Simple Games 224

8.7 Games with Coalition Structures 227

8.8 Games with a priori Unions 229

8.9 Multilinear Extensions of Games 234

8.11 A Summary of some Properties of the Main Solutions 238

9 Continuity Properties of Solutions 241

9.1 Upper Hemicontinuity of Solutions 241

9.2 Lower Hemicontinuity of Solutions 245

9.3 Continuity of the Prenucleolus 249

10 Dynamic Bargaining Procedures for the Kernel 251

10.1 Dynamic Systems for the Kernel and the Bargaining Set 252

10.2 Stable Sets of the Kernel and the Bargaining Set 259

10.3 Asymptotic Stability of the Nucleolus 262

Part II NTU Games

11 Cooperative Games in Strategic and Coalitional Form 269

11.1 Cooperative Games in Strategic Form 270

11.2 [alpha]- and [beta]-Effectiveness 272

11.3 Coalitional Games with Nontransferable Utility 277

11.4 Cooperative Games with Side Payments but without TU 278

12 The Core of NTU Games 281

12.1 Individual Rationality, Pareto Optimality, and the Core 282

12.2 Balanced NTU Games 284

12.3 Ordinal and Cardinal Convex Games 290

12.3.1 Ordinal Convex Games 290

12.3.2 Cardinal Convex Games 292

12.4 An Axiomatization of the Core 295

12.4.1 Reduced Games of NTU Games 296

12.4.2 Axioms for the Core 298

12.4.3 Proof of Theorem 12.4.8 299

12.5 Additional Properties and Characterizations 303

13 Some Values of NTU Games 309

13.1 The Shapley Value of NTU Games 309

13.2 A Characterization of the Shapley NTU Value 314

13.3 The Harsanyi Solution 320

13.4 The Consistent Shapley Value 325

14 Bargaining Sets for NTU Games 333

14.1 The Bargaining Set M 333

14.2 The Ordinal Bargaining Set M[superscript o] 337

14.3 A Proof of Billera's Theorem 342

14.4 Solutions Related to M[superscript o] 346

14.4.1 The Ordinal (Semi-) Reactive Bargaining Set 346

14.4.2 Solutions Related to the Prekernel 348.

Notes:

Includes bibliographical references (pages 357-369) and indexes.

ISBN:

1402074107

OCLC:

51983446