<div class="buying"><font size="3"><b class="sans">Introduction to the Theory of Cooperative Games (Theory and Decision Library C) (Hardcover)<!--Element not supported - Type: 8 Name: #comment--></b><br/></font>by <a href="http://www.amazon.com/exec/obidos/search-handle-url/002-5237687-7445669?%5Fencoding=UTF8&amp;search-type=ss&amp;index=books&amp;field-author=Bezalel%20Peleg"><font color="#003399">Bezalel Peleg</font></a> (Author), <a href="http://www.amazon.com/exec/obidos/search-handle-url/002-5237687-7445669?%5Fencoding=UTF8&amp;search-type=ss&amp;index=books&amp;field-author=Peter%20Sudh%C3%B6lter"><font color="#003399">Peter Sudh&ouml;lter</font></a> (Author) </div><div class="buying"><a href="http://www.amazon.com/gp/reader/3540729445/ref=sib_dp_pt/002-5237687-7445669#reader-link"><img id="prodImage" height="240" alt="Introduction to the Theory of Cooperative Games (Theory and Decision Library C)" src="http://ecx.images-amazon.com/images/I/41G0yB9v7vL._BO2,204,203,200_PIsitb-dp-500-arrow,TopRight,45,-64_OU01_AA240_SH20_.jpg" width="240" border="0"/></a></div><div class="buying"><li><b>Hardcover:</b> 328 pages </li><li><b>Publisher:</b> Springer; 2nd ed. edition (October 3, 2007) </li><li><b>Language:</b> English </li><li><div class="content"><b>Book Description</b><br/><p>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.</p></div></li><li><div class="content"><p> </p></div></li><li><div class="content"><p><strong><font size="4">Contents</font><br/>Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V</strong><br/>Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI<br/>List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII<br/>List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV<br/>Notation and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVII<br/><strong>1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br/></strong>1.1 Cooperative Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br/>1.2 Outline of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br/>1.2.1 TU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br/>1.2.2 NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br/>1.2.3 A Guide for the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br/>1.3 Special Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br/>1.3.1 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br/>1.3.2 Interpersonal Comparisons of Utility . . . . . . . . . . . . . . . . 5<br/>1.3.3 Nash’s Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br/><strong>Part I TU Games<br/>2 Coalitional TU Games and Solutions . . . . . . . . . . . . . . . . . . . . . 9</strong><br/>2.1 Coalitional Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br/>2.2 Some Families of Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br/>2.2.1 Market Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br/>2.2.2 Cost Allocation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br/>2.2.3 Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16<br/>2.3 Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br/>2.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26<br/><strong>3 The Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27</strong><br/>3.1 The Bondareva-Shapley Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 27<br/>3.2 An Application to Market Games . . . . . . . . . . . . . . . . . . . . . . . . . 32<br/>3.3 Totally Balanced Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34<br/>3.4 Some Families of Totally Balanced Games . . . . . . . . . . . . . . . . . 35<br/>3.4.1 Minimum Cost Spanning Tree Games . . . . . . . . . . . . . . . 35<br/>3.4.2 Permutation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br/>3.5 A Characterization of Convex Games . . . . . . . . . . . . . . . . . . . . . 39<br/>3.6 An Axiomatization of the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br/>3.7 An Axiomatization of the Core on Market Games . . . . . . . . . . 42<br/>3.8 The Core for Games with Various Coalition Structures . . . . . . 44<br/>3.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48<br/><strong>4 Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br/></strong>4.1 The Bargaining Set M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52<br/>4.2 Existence of the Bargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br/>4.3 Balanced Superadditive Games and the Bargaining Set . . . . . . 62<br/>4.4 Further Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64<br/>4.4.1 The Reactive and the Semi-reactive Bargaining Set . . . 65<br/>4.4.2 The Mas-Colell Bargaining Set . . . . . . . . . . . . . . . . . . . . . 69<br/>4.5 Non-monotonicity of Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . 72<br/>4.6 The Bargaining Set and Syndication: An Example . . . . . . . . . . 76<br/>4.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79<br/><strong>5 The Prekernel, Kernel, and Nucleolus . . . . . . . . . . . . . . . . . . . . 81<br/></strong>5.1 The Nucleolus and the Prenucleolus . . . . . . . . . . . . . . . . . . . . . . 82<br/>5.2 The Reduced Game Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86<br/>5.3 Desirability, Equal Treatment, and the Prekernel . . . . . . . . . . . 89<br/>5.4 An Axiomatization of the Prekernel . . . . . . . . . . . . . . . . . . . . . . . 91<br/>5.5 Individual Rationality and the Kernel . . . . . . . . . . . . . . . . . . . . . 94<br/>5.6 Reasonableness of the Prekernel and the Kernel . . . . . . . . . . . . 98<br/>5.7 The Prekernel of a Convex Game . . . . . . . . . . . . . . . . . . . . . . . . . 100<br/>5.8 The Prekernel and Syndication . . . . . . . . . . . . . . . . . . . . . . . . . . . 103<br/>5.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105<br/><strong>6 The Prenucleolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107<br/></strong>6.1 A Combinatorial Characterization of the Prenucleolus . . . . . . . 108<br/>6.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109<br/>6.3 An Axiomatization of the Prenucleolus . . . . . . . . . . . . . . . . . . . . 112<br/>6.3.1 An Axiomatization of the Nucleolus . . . . . . . . . . . . . . . . 115<br/>6.3.2 The Positive Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117<br/>6.4 The Prenucleolus of Games with Coalition Structures . . . . . . . 119<br/>6.5 The Nucleolus of Strong Weighted Majority Games . . . . . . . . . 120<br/>6.6 The Modiclus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124<br/>6.6.1 Constant-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129<br/>6.6.2 Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130<br/>6.6.3 Weighted Majority Games . . . . . . . . . . . . . . . . . . . . . . . . . 131<br/>6.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132<br/><strong>7 Geometric Properties of the ε-Core, Kernel, and Prekernel 133<br/></strong>7.1 Geometric Properties of the ε-Core . . . . . . . . . . . . . . . . . . . . . . . 133<br/>7.2 Some Properties of the Least-Core . . . . . . . . . . . . . . . . . . . . . . . . 136<br/>7.3 The Reasonable Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138<br/>7.4 Geometric Characterizations of the Prekernel and Kernel . . . . 142<br/>7.5 A Method for Computing the Prenucleolus . . . . . . . . . . . . . . . . 146<br/>7.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149<br/>X Contents<br/><strong>8 The Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151</strong><br/>8.1 Existence and Uniqueness of the Value . . . . . . . . . . . . . . . . . . . . 152<br/>8.2 Monotonicity Properties of Solutions and the Value . . . . . . . . . 156<br/>8.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159<br/>8.4 The Potential of the Shapley Value . . . . . . . . . . . . . . . . . . . . . . . 161<br/>8.5 A Reduced Game for the Shapley Value . . . . . . . . . . . . . . . . . . . 163<br/>8.6 The Shapley Value for Simple Games . . . . . . . . . . . . . . . . . . . . . 168<br/>8.7 Games with Coalition Structures . . . . . . . . . . . . . . . . . . . . . . . . . 170<br/>8.8 Games with A Priori Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172<br/>8.9 Multilinear Extensions of Games . . . . . . . . . . . . . . . . . . . . . . . . . 175<br/>8.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178<br/>8.11 A Summary of Some Properties of the Main Solutions . . . . . . . 179<br/><strong>9 Continuity Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 181</strong><br/>9.1 Upper Hemicontinuity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 181<br/>9.2 Lower Hemicontinuity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 184<br/>9.3 Continuity of the Prenucleolus . . . . . . . . . . . . . . . . . . . . . . . . . . . 187<br/>9.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188<br/><strong>10 Dynamic Bargaining Procedures for the Kernel and the<br/>Bargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189</strong><br/>10.1 Dynamic Systems for the Kernel and the Bargaining Set . . . . . 190<br/>10.2 Stable Sets of the Kernel and the Bargaining Set . . . . . . . . . . . 195<br/>10.3 Asymptotic Stability of the Nucleolus . . . . . . . . . . . . . . . . . . . . . 198<br/>10.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199<br/>Part II NTU Games<br/><strong>11 Cooperative Games in Strategic and Coalitional Form . . . . 203</strong><br/>11.1 Cooperative Games in Strategic Form . . . . . . . . . . . . . . . . . . . . . 203<br/>11.2 α- and β-Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205<br/>11.3 Coalitional Games with Nontransferable Utility . . . . . . . . . . . . 209<br/>Contents XI<br/>11.4 Cooperative Games with Side Payments but Without<br/>Transferable Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210<br/>11.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212<br/><strong>12 The Core of NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213</strong><br/>12.1 Individual Rationality, Pareto Optimality, and the Core . . . . . 214<br/>12.2 Balanced NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215<br/>12.3 Ordinal and Cardinal Convex Games. . . . . . . . . . . . . . . . . . . . . . 220<br/>12.3.1 Ordinal Convex Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . 220<br/>12.3.2 Cardinal Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 222<br/>12.4 An Axiomatization of the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 224<br/>12.4.1 Reduced Games of NTU Games . . . . . . . . . . . . . . . . . . . . 224<br/>12.4.2 Axioms for the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226<br/>12.4.3 Proof of Theorem 12.4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 227<br/>12.5 Additional Properties and Characterizations . . . . . . . . . . . . . . . 230<br/>12.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233<br/><strong>13 The Shapley NTU Value and the Harsanyi Solution . . . . . . 235</strong><br/>13.1 The Shapley Value of NTU Games. . . . . . . . . . . . . . . . . . . . . . . . 235<br/>13.2 A Characterization of the Shapley NTU Value . . . . . . . . . . . . . 239<br/>13.3 The Harsanyi Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243<br/>13.4 A Characterization of the Harsanyi Solution . . . . . . . . . . . . . . . 247<br/>13.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251<br/><strong>14 The Consistent Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253<br/></strong>14.1 For Hyperplane Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253<br/>14.2 For p-Smooth Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257<br/>14.3 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261<br/>14.3.1 The Role of IIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264<br/>14.3.2 Logical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265<br/>14.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267<br/>XII Contents<br/><strong>15 On the Classical Bargaining Set and the Mas-Colell<br/>Bargaining Set for NTU Games . . . . . . . . . . . . . . . . . . . . . . . . . . 269</strong><br/>15.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270<br/>15.1.1 The Bargaining Set M. . . . . . . . . . . . . . . . . . . . . . . . . . . . 270<br/>15.1.2 The Mas-Colell Bargaining Set MB and Majority<br/>Voting Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272<br/>15.1.3 The 3 × 3 Voting Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 274<br/>15.2 Voting Games with an Empty Mas-Colell Bargaining Set . . . . 278<br/>15.3 Non-levelled NTU Games with an Empty Mas-Colell<br/>Prebargaining Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282<br/>15.3.1 The Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283<br/>15.3.2 Non-levelled Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286<br/>15.4 Existence Results for Many Voters . . . . . . . . . . . . . . . . . . . . . . . . 289<br/>15.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292<br/><strong>16 Variants of the Davis-Maschler Bargaining Set for NTU<br/>Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295</strong><br/>16.1 The Ordinal Bargaining Set Mo . . . . . . . . . . . . . . . . . . . . . . . . . . 295<br/>16.2 A Proof of Billera’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299<br/>16.3 Solutions Related to Mo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302<br/>16.3.1 The Ordinal Reactive and the Ordinal Semi-Reactive<br/>Bargaining Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302<br/>16.3.2 Solutions Related to the Prekernel . . . . . . . . . . . . . . . . . . 303<br/>16.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308<br/><strong>References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311<br/>Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321<br/>Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323</strong></p></div></li></div>