An Introductory Course on Mathematical Game Theory and Applications, 2nd Edition

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 psychology, computer science, artificial intelligence, biology, and political 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. In this second edition, the content on cooperative games is considerably strengthened, with a new chapter on applications of cooperative games and operations research, including some material on computational aspects and applications outside academia. This book is published in cooperation with Real Sociedád Matematica Española. Cover 1 Table of Contents 8 Preface 12 Chapter 1. Introduction to Decision Theory 16 1.1. Preliminaries 16 1.2. Ordinal Utility 18 1.3. Linear Utility 21 Chapter 2. Strategic Games 28 2.1. Introduction to Strategic Games 28 2.2. Nash Equilibrium in Strategic Games 33 2.3. Two-Player Zero-Sum Games 39 2.4. Mixed Strategies in Finite Games 43 2.5. Bimatrix Games 46 2.6. Matrix Games 52 2.7. Algorithms for Matrix Games 58 2.8. Matrix Games and Linear Programming 67 2.9. Refinements of Nash Equilibrium in Finite Games 74 2.10. A Basic Model of Knowledge 88 2.11. Correlated Equilibrium 91 2.12. On the Epistemic Foundations of the Different Solution Concepts for Strategic Games 96 2.13. Fixed-Point Theorems 103 2.14. Extreme Points and Convex Sets: Krein–Milman Theorem 107 Exercises of Chapter 2 110 Chapter 3. Extensive Games 116 3.1. Introduction to Extensive Games 116 3.2. Strategies in Extensive Games: Mixed Strategies vs. Behavior Strategies 121 3.3. Nash Equilibrium in Extensive Games 127 3.4. Subgame Perfect Equilibrium 134 3.5. Sequential Equilibrium 142 3.6. Further Refinements 149 3.7. Repeated Games 160 Exercises of Chapter 3 175 Chapter 4. Games with Incomplete Information 180 4.1. Incomplete Information: Introduction and Modeling 180 4.2. Bayesian Games and Bayesian Nash Equilibrium 182 4.3. The Chain Store Paradox in Perspective 188 4.4. A First Application of Bayesian Games: Auctions 195 4.5. A Second Application of Bayesian Games: Mechanism Design and the Revelation Principle 203 4.6. Extensive Games with Incomplete Information: Multistage Games and Perfect Bayesian Equilibrium 207 4.7. An Outline of Harsanyi's Approach 214 Exercises of Chapter 4 217 Chapter 5. Fundamentals of Cooperative Games 220 5.1. Introduction to Cooperative Games 220 5.2. Nontransferable Utility Games 221 5.3. Bargaining 223 5.4. Transferable Utility Games 231 5.5. The Core and Related Concepts 234 5.6. The Shapley Value 243 5.7. Computation of the Shapley Value 248 5.8. The Banzhaf Value 254 5.9. Equal Division Values 257 5.10. The Nucleolus 260 5.11. The Core-Center 264 5.12. Convex Games 268 5.13. Games with Communication Graphs: The Myerson Value 271 5.14. Games with A Priori Unions: The Owen Value 276 5.15. Noncooperative Models in Cooperative Game Theory: Implementation Theory 280 Exercises of Chapter 5 298 Chapter 6. Applications of Cooperative Games 300 6.1. Preliminaries 300 6.2. Airport Problems and Airport Games 301 6.3. Bankruptcy Problems and Bankruptcy Games 308 6.4. Voting Problems and Voting Games: Power Indices 318 6.5. Apportionments and Data Science 330 6.6. Linear Production Games 335 6.7. Maximum Flow Games 339 6.8. Inventory Games 342 6.9. Project Planning 348 6.10. Assignment Games 354 6.11. Cooperation on Minimum Cost Spanning Tree Problems 362 6.12. Cooperation on Queueing Problems 373 6.13. Cooperation on Sequencing Problems 378 Exercises of Chapter 6 382 Bibliography 386 Notations 410 Index of Authors 412 Index of Solution Concepts 418 Subject Index 420 Back Cover 432