Introduction to the theory of cooperative games:
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| Main Authors: | , |
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
[2007]
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| Edition: | Second edition |
| Series: | Theory and decision library. Series C, Game theory, mathematical programming and operations research
volume 34 |
| Subjects: | |
| Online Access: | BTU01 FHM01 FHR01 FNU01 UBG01 UBR01 UBT01 UBY01 UEI03 UPA01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XIX, 328 Seiten) |
| ISBN: | 9783540729457 |
| DOI: | 10.1007/978-3-540-72945-7 |
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| 245 | 1 | 0 | |a Introduction to the theory of cooperative games |c Bezalel Peleg, Peter Sudhölter |
| 250 | |a Second edition | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c [2007] | |
| 264 | 4 | |c © 2007 | |
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Record in the Search Index
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| adam_text | 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
VIII Contents
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 AI 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
Contents IX
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 Geometrie Properties of the e Core, Kernel, and Prekernel 133
7.1 Geometrie Properties of the e Core 133
7.2 Some Properties of the Least Core 136
7.3 The Reasonable Set 138
7.4 Geometrie 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
811 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 Stahle 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 Strategie and Coalitional Form ... 203
11.1 Cooperative Games in Strategie Form 203
11.2 a and /3 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 3x3 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
List of Figures
Fig. 2.2.1 Connection Cost 15
Fig. 4.4.1 The Projective Seven Person Game 66
Fig. 13.1.1 The Shapley Value 237
List of Tables
Table 8.11.1 Solutions and Properties 179
Table 15.1.1 Preference Profile of a 4 Person Voting Problem 274
Table 15.1.2 Preference Profile of the 3 x 3 Voting Paradox 275
Table 15.1.3 Preference Profile of a 4 Alternative Voting Problem ... 276
Table 15.2.4 Preference Profile leading to an empty VMB 278
Table 15.3.5 Preference Profile on 10 Alternatives 283
Table 15.3.6 Domination Relation 283
Table 15.3.7 Constructions of Streng Objections 284
|
| adam_txt |
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
VIII Contents
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 AI 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
Contents IX
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 Geometrie Properties of the e Core, Kernel, and Prekernel 133
7.1 Geometrie Properties of the e Core 133
7.2 Some Properties of the Least Core 136
7.3 The Reasonable Set 138
7.4 Geometrie 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
811 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 Stahle 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 Strategie and Coalitional Form . 203
11.1 Cooperative Games in Strategie Form 203
11.2 a and /3 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 3x3 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
List of Figures
Fig. 2.2.1 Connection Cost 15
Fig. 4.4.1 The Projective Seven Person Game 66
Fig. 13.1.1 The Shapley Value 237
List of Tables
Table 8.11.1 Solutions and Properties 179
Table 15.1.1 Preference Profile of a 4 Person Voting Problem 274
Table 15.1.2 Preference Profile of the 3 x 3 Voting Paradox 275
Table 15.1.3 Preference Profile of a 4 Alternative Voting Problem . 276
Table 15.2.4 Preference Profile leading to an empty VMB 278
Table 15.3.5 Preference Profile on 10 Alternatives 283
Table 15.3.6 Domination Relation 283
Table 15.3.7 Constructions of Streng Objections 284 |
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| author | Peleg, Bezalel 1936- Sudhölter, Peter |
| author_GND | (DE-588)170089606 (DE-588)170772306 |
| author_facet | Peleg, Bezalel 1936- Sudhölter, Peter |
| author_role | aut aut |
| author_sort | Peleg, Bezalel 1936- |
| author_variant | b p bp p s ps |
| building | Verbundindex |
| bvnumber | BV022718809 |
| classification_rvk | QH 430 SK 860 |
| collection | ZDB-2-SBE |
| ctrlnum | (OCoLC)611868600 (DE-599)BVBBV022718809 |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| doi_str_mv | 10.1007/978-3-540-72945-7 |
| edition | Second edition |
| format | Electronic eBook |
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| id | DE-604.BV022718809 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:29:21Z |
| indexdate | 2024-07-09T21:04:25Z |
| institution | BVB |
| isbn | 9783540729457 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015924556 |
| oclc_num | 611868600 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG DE-945 DE-739 DE-706 DE-898 DE-BY-UBR DE-M347 DE-1049 DE-355 DE-BY-UBR DE-634 DE-703 |
| owner_facet | DE-473 DE-BY-UBG DE-945 DE-739 DE-706 DE-898 DE-BY-UBR DE-M347 DE-1049 DE-355 DE-BY-UBR DE-634 DE-703 |
| physical | 1 Online-Ressource (XIX, 328 Seiten) |
| psigel | ZDB-2-SBE |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Theory and decision library. Series C, Game theory, mathematical programming and operations research |
| series2 | Theory and decision library. Series C, Game theory, mathematical programming and operations research |
| spelling | Peleg, Bezalel 1936- Verfasser (DE-588)170089606 aut Introduction to the theory of cooperative games Bezalel Peleg, Peter Sudhölter Second edition Berlin ; Heidelberg ; New York Springer [2007] © 2007 1 Online-Ressource (XIX, 328 Seiten) txt rdacontent c rdamedia cr rdacarrier Theory and decision library. Series C, Game theory, mathematical programming and operations research volume 34 Kooperatives Spiel (DE-588)4120603-4 gnd rswk-swf Spieltheorie (DE-588)4056243-8 gnd rswk-swf Spieltheorie (DE-588)4056243-8 s Kooperatives Spiel (DE-588)4120603-4 s b DE-604 Sudhölter, Peter Verfasser (DE-588)170772306 aut Erscheint auch als Druck-Ausgabe 978-3-540-72944-0 Theory and decision library. Series C, Game theory, mathematical programming and operations research volume 34 (DE-604)BV044969009 34 https://doi.org/10.1007/978-3-540-72945-7 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015924556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Peleg, Bezalel 1936- Sudhölter, Peter Introduction to the theory of cooperative games Theory and decision library. Series C, Game theory, mathematical programming and operations research Kooperatives Spiel (DE-588)4120603-4 gnd Spieltheorie (DE-588)4056243-8 gnd |
| subject_GND | (DE-588)4120603-4 (DE-588)4056243-8 |
| title | Introduction to the theory of cooperative games |
| title_auth | Introduction to the theory of cooperative games |
| title_exact_search | Introduction to the theory of cooperative games |
| title_exact_search_txtP | Introduction to the theory of cooperative games |
| title_full | Introduction to the theory of cooperative games Bezalel Peleg, Peter Sudhölter |
| title_fullStr | Introduction to the theory of cooperative games Bezalel Peleg, Peter Sudhölter |
| title_full_unstemmed | Introduction to the theory of cooperative games Bezalel Peleg, Peter Sudhölter |
| title_short | Introduction to the theory of cooperative games |
| title_sort | introduction to the theory of cooperative games |
| topic | Kooperatives Spiel (DE-588)4120603-4 gnd Spieltheorie (DE-588)4056243-8 gnd |
| topic_facet | Kooperatives Spiel Spieltheorie |
| url | https://doi.org/10.1007/978-3-540-72945-7 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015924556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044969009 |
| work_keys_str_mv | AT pelegbezalel introductiontothetheoryofcooperativegames AT sudholterpeter introductiontothetheoryofcooperativegames |