Game Theory: An Introduction
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A fundamental introduction to modern game theory from a mathematical viewpoint Game theory arises in almost every fact of human and inhuman interaction since oftentimes during these communications objectives are opposed or cooperation is viewed as an option. From economics and finance to biology and computer science, researchers and practitioners are often put in complex decision-making scenarios, whether they are interacting with each other or working with evolving technology and artificial intelligence. Acknowledging the role of mathematics in making logical and advantageous decisions, Game Theory: An Introduction uses modern software applications to create, analyze, and implement effective decision-making models. While most books on modern game theory are either too abstract or too applied, this book provides a balanced treatment of the subject that is both conceptual and hands-on. Game Theory introduces readers to the basic theories behind games and presents real-world examples from various fields of study such as economics, political science, military science, finance, biological science as well as general game playing. A unique feature of this book is the use of Maple to find the values and strategies of games, and in addition, it aids in the implementation of algorithms for the solution or visualization of game concepts. Maple is also utilized to facilitate a visual learning environment of game theory and acts as the primary tool for the calculation of complex non-cooperative and cooperative games. Important game theory topics are presented within the following five main areas of coverage: Two-person zero sum matrix games Nonzerosum games and the reduction to nonlinear programming Cooperative games, including discussion of both the Nucleolus concept and the Shapley value Bargaining, including threat strategies Evolutionary stable strategies and population games Although some mathematical competence is assumed, appendices are provided to act as a refresher of the basic concepts of linear algebra, probability, and statistics. Exercises are included at the end of each section along with algorithms for the solution of the games to help readers master the presented information. Also, explicit Maple and Mathematica® commands are included in the book and are available as worksheets via the book's related Web site. The use of this software allows readers to solve many more advanced and interesting games without spending time on the theory of linear and nonlinear programming or performing other complex calculations. With extensive examples illustrating game theory's wide range of relevance, this classroom-tested book is ideal for game theory courses in mathematics, engineering, operations research, computer science, and economics at the upper-undergraduate level. It is also an ideal companion for anyone who is interested in the applications of game theory.
Preface xiAcknowledgments xvIntroduction xviiMatrix Two-Person Games 1The Basics 1Problems 14The von Neumann Minimax Theorem 15Problems 21Mixed Strategies 22Dominated Strategies 34Solving 2 x 2 Games Graphically 37Problems 38Graphical Solution of 2 x m and n x 2 Games 39Problems 46Best Response Strategies 47Problems 53Solution Methods for Matrix Games 55Solution of Some Special Games 552 x 2 Games Revisited 55Problems 59Invertible Matrix Games 60Problems 65Symmetric Games 68Problems 72Matrix Games and Linear Programming 73A Direct Formulation Without Transforming: Method 2 81Problems 87Linear Programming and the Simplex Method (Optional) 90The Simplex Method Step by Step 93Problems 100A Game Theory Model of EconomicGrowth (Optional) 101Problems 107Two-Person Nonzero Sum Games 109The Basics 109Problems 1172 x 2 Bimatrix Games 118Problems 128Interior Mixed Nash Points by Calculus 129Problems 139Proof that There Is a Nash Equilibrium for Bimatrix Games (Optional) 141Nonlinear Programming Method for Nonzero Sum Two-Person Games 143Problems 153Choosing Among Several Nash Equilibria (Optional) 155Problems 160N-Person Nonzero Sum Games with a Continuum of Strategies 163The Basics 163Economics Applications of Nash equilibria 169Problems 184Duels (optional) 186Problems 193Auctions (Optional) 195Complete Information 200Problems 201Incomplete Information 202Symmetric Independent Private Value Auctions 205Problems 217Cooperative Games 219Coalitions and Characteristic Functions 219Problems 240Finding the Least Core 241Problems 248The Nucleolus 249Problems 263The Shapley Value 264Problems 276Bargaining 277The Nash Model with Security Point 285Threats 292Problems 305Evolutionary Stable Strategies and Population Games 309Evolution 310Problems 319Population Games 320Problems 340The Essentials of Matrix Analysis 345The Essentials of Probability 351The Essentials of Maple 357The Mathematica Commands 363Biographies 379Problem Solutions 383References 409Index 411
