Game Theory (Second Edition)

Game theory is a branch of modern applied mathematics that aims to analyse various problems of conflict between parties that have opposed similar or simply different interests. Games are grouped into several classes according to some important features. In Game Theory (2nd Edition), Petrosyan and Zenkevich consider zero-sum two-person games, strategic N-person games in normal form, cooperative games, games in extensive form with complete and incomplete information, differential pursuit games and differential cooperative, and non-cooperative N-person games. The 2nd edition updates heavily from the 1st edition published in 1996. Contents 10 Preface 6 Acknowledgments 8 1 Matrix Games 14 1.1 Definition of a Two-Person Zero-Sum Game in Normal Form 14 1.2 Maximin andMinimax Strategies 20 1.3 Saddle Points 23 1.4 Mixed Extension of a Game 30 1.5 Convex Sets and Systems of Linear Inequalities 35 1.6 Existence of a Solution of the Matrix Game in Mixed Strategies 40 1.7 Properties of Optimal Strategies and Value of the Game 45 1.8 Dominance of Strategies 57 1.9 Completely Mixed and Symmetric Games 65 1.10 Iterative Methods of Solving Matrix Games 72 1.11 Exercises and Problems 78 2 Infinite Zero-Sum Two-Person Games 84 2.1 Infinite Games 84 2.2 -Saddle Points, -Optimal Strategies 88 2.3 Mixed Strategies 95 2.4 Games with Continuous Payoff Functions 105 2.5 Games with a Convex Payoff Function 114 2.6 Simultaneous Games of Pursuit 128 2.7 One Class of Games with a Discontinuous Payoff Function 136 2.8 Infinite Simultaneous Search Games 140 2.9 A Poker Model 147 2.10 Exercises and Problems 173 3 Nonzero-Sum Games 178 3.1 Definition of Noncooperative Game in Normal Form 178 3.2 Optimality Principles in Noncooperative Games 184 3.3 Mixed Extension of Noncooperative Game 198 3.4 Existence of Nash Equilibrium 204 3.5 Kakutani Fixed-Point Theorem and Proof of Existence of an Equilibrium in n-Person Games 210 3.6 Refinements of Nash Equilibria 215 3.7 Properties of Optimal Solutions 220 3.8 Symmetric Bimatrix Games and Evolutionary Stable Strategies 226 3.9 Equilibrium in Joint Mixed Strategies 231 3.10 The Bargaining Problem 236 3.11 Exercises and Problems 247 4 Cooperative Games 254 4.1 Games in Characteristic Function Form 254 4.2 The Core and NM-Solution 266 4.3 The Shapley Value 278 4.4 The Potential of the Shapley Value 287 4.5 The τ-Value and Nucleolus 292 4.6 Exercises and Problems 295 5 Positional Games 300 5.1 Multistage Games with Perfect Information 300 5.2 Absolute Equilibrium (Subgame-Perfect) 308 5.3 Fundamental Functional Equations 319 5.4 Penalty Strategies 322 5.5 Repeated Games and Equilibrium in Punishment (Penalty) Strategies 326 5.6 Hierarchical Games 328 5.7 Hierarchical Games (Cooperative Version) 332 5.8 Multistage Games with Incomplete Information 341 5.9 Behavior Strategy 350 5.10 Functional Equations for Simultaneous Multistage Games 360 5.11 Cooperative Multistage Games with Perfect Information 371 5.12 One-Way Flow Two-Stage Network Games 390 5.13 Exercises and Problems 404 6 N-Person Differential Games 414 6.1 Optimal Control Problem 414 6.2 Differential Games and Their Solution Concepts 425 6.3 Application of Differential Games in Economics 434 6.4 Infinite-Horizon Differential Games 437 6.5 Cooperative Differential Games in Characteristic Function Form 444 6.6 Imputation in a Dynamic Context 449 6.7 Principle of Dynamic Stability 452 6.8 Dynamic Stable Solutions 453 6.9 Payoff Distribution Procedure 454 6.10 An Analysis in Pollution Control 458 6.11 Illustration with Specific Functional Forms 467 6.12 Exercises and Problems 472 7 Zero-Sum Differential Games 476 7.1 Differential Zero-Sum Games with Prescribed Duration 476 7.2 Multistage Perfect-Information Games with an Infinite Number of Alternatives 489 7.3 Existence of -Equilibria in Differential Games with Prescribed Duration 495 7.4 Differential Time-Optimal Games of Pursuit 503 7.5 Necessary and Sufficient Condition for Existence of Optimal Open-Loop Strategy for Evader 512 7.6 Fundamental Equation 517 7.7 Methods of Successive Approximations for Solving Differential Games of Pursuit 527 7.8 Examples of Solutions to Differential Games of Pursuit 532 7.9 Games of Pursuit with Delayed Information for Pursuer 538 7.10 Exercises and Problems 547 Bibliography 556 Index 562 Game theory is a branch of modern applied mathematics that aims to analyze various problems of conflict between parties that have opposed, similar or simply different interests. Games are grouped into several classes according to some important features. In this volume zero-sum two-person games, strategic n-person games in normal form, cooperative games, games in extensive form with complete and incomplete information, differential pursuit games and differential cooperative n-person games are considered