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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.

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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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