Chapters in Game Theory (eBook)

Preface 6
About Stef Tijs 7
About this book 7
About the authors 10
Contents 14
Chapter 1 Stochastic Cooperative Games: Theory and Applications 19
1.1 Introduction 19
1.2 Cooperative Decision-Making under Risk 23
1.2.1 Chance-Constrained Games 23
1.2.2 Stochastic Cooperative Games with Transfer Payments 25
1.2.3 Stochastic Cooperative Games without Transfer Payments 29
1.3 Cost Allocation in a Network Tree 33
1.4 Bankruptcy Problems with Random Estate 37
1.5 Concluding Remarks 40
Appendix 41
References 43
Chapter 2 Sequencing Games: a Survey 45
2.1 Introduction 45
2.2 Games Related to Sequencing Games 47
2.3 Sequencing Situations and Sequencing Games 49
2.4 On Sequencing Games with Ready Times or Due Dates 54
2.5 On Sequencing Games with Multiple Machines 58
2.6 On Sequencing Games with more Admissible Rearrangements 63
References 66
Chapter 3 Game Theory and the Market 69
3.1 Introduction 69
3.2 Von Neumann, Morgenstern and Nash 70
3.3 Bargaining 75
3.4 Markets 79
3.5 Auctions 87
3.6 Conclusion 95
References 96
Chapter 4 On the Number of Extreme Points of the Core of a Transferable Utility Game 101
4.1 Introduction 101
4.2 Main Results 103
4.3 The Core of a Transferable Utility Game 106
4.4 Strict Exact Games 109
4.5 Concluding Remarks 112
References 114
Chapter 5 Consistency and Potentials in Cooperative TU- Games: Sobolev’s Reduced Game Revived 117
5.1 Introduction 117
5.2 Consistency Property for Solutions that Admit a Potential 120
5.3 Consistency Property for Pseudovalues: a Detailed Exposition 126
5.4 Concluding remarks 134
5.5 Two technical proofs 134
References 137
Chapter 6 On the Set of Equilibria of a Bimatrix Game: a Survey 139
6.1 Introduction 139
6.2 Bimatrix Games and Equilibria 142
6.3 Some Observations by Nash 142
6.4 The Approach of Vorobev and Kuhn 144
6.5 The Approach of Mangasarian and Winkels 147
6.7 The Approach of Jansen 151
6.8 The Approach of Quintas 154
6.9 The Approach of Jurg and Jansen 154
6.10 The Approach of Vermeulen and Jansen 158
References 159
Chapter 7 Concave and Convex Serial Cost Sharing 161
7.1 Introduction 161
7.2 The Cost Sharing Model 162
7.3 The Convex and the Concave Serial Cost Sharing Rule 164
References 172
Chapter 8 Centrality Orderings in Social Networks 175
8.1 Introduction 175
8.2 Examples of Centrality Orderings 177
8.3 Cover Centrality Ordering 182
8.4 Degree Centrality Ordering 186
8.5 Median Centrality Ordering 191
8.6 Independence of the Characterizing Conditions 195
References 197
Chapter 9 The Shapley Transfer Procedure for NTU- Games 201
9.1 Introduction 201
9.2 Main Concepts 203
9.3 Nonemptiness of Transfer Solutions 207
9.4 A Characterization 210
9.5 Applications 213
9.5.1 The Shapley Value 213
9.5.2 The Core 214
9.5.3 The Nucleolus 216
9.5.4 The t-Value 217
9.6 Concluding Remarks 220
References 220
Chapter 10 The Nucleolus as Equilibrium Price 223
10.1 Introduction 223
10.2 Preliminaries 225
10.2.1 Economies with Indivisible Goods and Money 226
10.2.2 Preliminaries about TU-Games 227
10.3 Stable Equilibria 228
10.4 The Existence of Price Equilibria: Necessary and Sufficient Conditions 234
10.5 The Nucleolus as Regular Price Vector 236
References 240
Chapter 11 Network Formation, Costs, and Potential Games 241
11.1 Introduction 241
11.2 Literature Review 242
11.3 Network Formation Model in Strategic Form 246
11.4 Potential Games 251
11.5 Potential Maximizer 256
References 262
Chapter 12 Contributions to the Theory of Stochastic Games 265
12.1 The Stochastic Game Model 265
12.2 Zero-Sum Stochastic Games 268
12.3 General-Sum Stochastic Games 273
References 279
Chapter 13 Linear (Semi-) Infinite Programs and Cooperative Games 285
13.1 Introduction 285
13.2 Semi-infinite Programs and Games 286
13.2.1 Flow games 286
13.2.2 Linear Production Games 288
13.2.3 Games Involving Linear Transformation of Products 291
13.3 Infinite Programs and Games 294
13.3.1 Assignment Games 294
13.3.2 Transportation Games 297
13.4 Concluding remarks 301
References 302
Chapter 14 Population Uncertainty and Equilibrium Selection: a Maximum Likelihood Approach 305
14.1 Introduction 305
14.2 Preliminaries 307
14.2.1 Topology 307
14.2.2 Measure Theory 308
14.2.3 Game Theory 309
14.3 Games with Population Uncertainty 310
14.4 Maximum Likelihood Equilibria 311
14.5 Measurability 315
14.6 Random Action Sets 317
14.8 Robustness Against Randomization 320
14.9 Weakly Strict Equilibria 323
14.10 Approximate Maximum Likelihood Equilibria 326
References 331