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Introduction to the theory of cooperative games

Author: Peleg, Bezalel ; Sudhölter, Peter Series: Theory and decision library ; 34 Publisher: Springer, 2007.Edition: 2nd ed.Language: EnglishDescription: 328 p. ; 24 cm.ISBN: 9783540729440Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print HB144 .P45 2007
(Browse shelf)
001252354
Available 001252354
Total holds: 0

Includes bibliographical references and index

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Introduction To The Theory of Cooperative Games 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 e-Core, Kernel, and Prekernel 133 7.1 Geometric Properties of the r-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 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 a- and 3-Effectiveness ............................................................... 205 11.3 Coalitional Games with Nontransferable Utility ......................... 209 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 HA............................................................... 264 14.3.2 Logical Independence .................................................... 265 14.4 Notes and Comments ............................................................... 267 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 x 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 M°.................................................. 295 16.2 A Proof of Billera's Theorem...................................................... 299 16.3 Solutions Related to M° ............................................................ 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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