Game theory provides a mathematical setting for analyzing competition and cooperation in interactive situations. The theory has been famously applied in economics, but is relevant in many other sciences, such as political science, biology, and, more recently, computer science. This book presents an introductory and up-to-date course on game theory addressed to mathematicians and economists, and to other scientists having a basic mathematical background. The book is self-contained, providing a formal description of the classic game-theoretic concepts together with rigorous proofs of the main results in the field. The theory is illustrated through abundant examples, applications, and exercises. The style is distinctively concise, while offering motivations and interpretations of the theory to make the book accessible to a wide readership. The basic concepts and results of game theory are given a formal treatment, and the mathematical tools necessary to develop them are carefully presented. Cooperative games are explained in detail, with bargaining and TU-games being treated as part of a general framework. The authors stress the relation between game theory and operations research. The book is suitable for a graduate or an advanced undergraduate course on game theory. Table of Contents: Introduction to decision theory; Strategic games; Extensive games; Games with incomplete information; Cooperative games; Bibliography; Notations; Index of authors; Index of solution concepts; Subject index. (GSM/115)
Author(s): Julio Gonzalez-Diaz, Ignacio Garcia-Jurado, M. Gloria Fiestras-Janeiro
Series: Graduate Studies in Mathematics 115
Publisher: American Mathematical Society
Year: 2010
Language: English
Pages: 342
Preface xi
Chapter 1. Introduction to Decision Theory 1
§1.1. Preliminaries 1
§1.2. Ordinal Utility 3
§1.3. Linear Utility 6
Chapter 2. Strategic Games 13
§2.1. Introduction to Strategic Games 13
§2.2. Nash Equilibrium in Strategic Games 18
§2.3. Two-Player Zero-Sum Games 24
§2.4. Mixed Strategies in Finite Games 28
§2.5. Bimatrix Games 31
§2.6. Matrix Games 37
§2.7. Algorithms for Matrix Games 43
§2.8. Matrix Games and Linear Programming 52
§2.9. Refinements of Nash Equilibrium in Finite Games 59
§2.10. A Basic Model of Knowledge 72
§2.11. Correlated Equilibrium 75
§2.12. On the Epistemic Foundations of the Different Solution Concepts for Strategic Games 80
§2.13. Fixed-Point Theorems 87
§2.14. On Extreme Points and Convex Sets: Krein-Milman Theorem 91
Exercises of Chapter 2 94
Chapter 3. Extensive Games 99
§3.1. Introduction to Extensive Games 99
§3.2. Strategies in Extensive Games: Mixed Strategies vs. Behavior Strategies 104
§3.3. Nash Equilibrium in Extensive Games 110
§3.4. Subgame Perfect Equilibrium 117
§3.5. Sequential Equilibrium 124
§3.6. Further Refinements 132
§3.7. Repeated Games 146
Exercises of Chapter 3 159
Chapter 4. Games with Incomplete Information 163
§4.1. Incomplete Information: Introduction and Modeling 163
§4.2. Bayesian Games and Bayesian Nash Equilibrium 165
§4.3. The Chain Store Paradox in Perspective 171
§4.4. A First Application of Bayesian Games: Auctions 178
§4.5. A Second Application of Bayesian Games: Mechanism Design and the Revelation Principle 186
§4.6. Extensive Games with Incomplete Information: Multistage Games and Perfect Bayesian Equilibrium 190
§4.7. An Outline of Harsanyi's Approach 197
Exercises of Chapter 4 200
Chapter 5. Cooperative Games 203
§5.1. Introduction to Cooperative Games 203
§5.2. Nontransferable Utility Games 204
§5.3. Bargaining 206
§5.4. Transferable Utility Games 214
5.5. The Core and Related Concepts 217
§5.6. The Shapley Value 226
§5.7. The Nucleolus 231
§5.8. Convex Games 235
§5.9. Noncooperative Models in Cooperative Game Theory: Implementation Theory 238
§5.10. Airport Problems and Airport Games 256
§5.11. Bankruptcy Problems and Bankruptcy Games 261
§5.12. Voting Problems and Voting Games: Power Indices 270
§5.13. Cooperation in Operations Research Models 275
Exercises of Chapter 5 289
Bibliography 293
Notations 309
Index of Authors 311
Index of Solution Concepts 315
Subject Index 317
