One of the definitive works in game theory, this fascinating volume offers an original look at methods of obtaining solutions for conflict situations. Combining the principles of game theory, the calculus of variations, and control theory, the author considers and solves an amazing array of problems: military, pursuit and evasion, games of firing and maneuver, athletic contests, and many other problems of conflict.
Beginning with general definitions and the basic mathematics behind differential game theory, the author proceeds to examinations of increasingly specific techniques and applications: dispersal, universal, and equivocal surfaces; the role of game theory in warfare; development of an effective theory despite incomplete information; and more. All problems and solutions receive clearly worded, illuminating discussions, including detailed examples and numerous formal calculations.
The product of fifteen years of research by a highly experienced mathematician and engineer, this volume will acquaint students of game theory with practical solutions to an extraordinary range of intriguing problems.
CHAPTER 1 An Introduction 1.1 The Theory of Games 1.2 The State of Control Variables 1.3 Battle Games 1.4 Games with Moving Craft 1.5 Pursuit Games 1.6 Games of Kind and Games of Degrees 1.7 Strategies 1.8 "Dogfights, Firing Games, Programming, and Athletics" 1.9 Two Examples 1.10 A Perspective on Precision 1.11 A Perspective on Progress 1.12 On Reading This Book "CHAPTER 2 Definitions, Formulation, and Assumptions" 2.1 The Kinematic Situation 2.2 The Realistic and Reduced Space 2.3 Termination of the Game 2.4 The Payoff 2.5 Games of Kind and Games of Degree 2.6 Strategies 2.7 Canonization of the Vectograms 2.8 A Lemma on Circular Vectograms CHAPTER 3 Discrete Differential Games 3.1 Introduction 3.2 The General Discrete Game 3.3 Battles of Extinction 3.4 Two Discrete Pursuit Games 3.5 Quasi-Discrete Games CHAPTER 4 The Basic Mathematics and the Solution Technique in the Small 4.1 The Nature of a Solution 4.2 The Main Equation 4.3 Semipermeable Surfaces and a Second Derivation of the Main Equation 4.4 The Verification Theorem 4.5 The Path Equations 4.6 The Retrogression Principle 4.7 The Initial Conditions CHAPTER 5 Mainly Examples: Transition Surfaces: Integral Constraints 5.1 Transition Surfaces 5.2 The Dolichobrachistochrone 5.3 The Relationship to the Euler Equation 5.4 The War of Attrition and Attack 5.5 The Isotropic Rocket Pursuit Game 5.6 An Optimal Program of Steel Production 5.7 Integral Constraints CHAPTER 6 Efferent of Dispersal Surfaces 6.1 Singular Surfaces 6.2 Dispersal Surfaces 6.3 The Nature of Dispersal Surfaces 6.4 The Question of the Perpetuated Dilemma 6.5 The Construction of Dispersal Surfaces 6.6 Further Examples 6.7 The Geometric Method for Simple Pursuit Games of Kind 6.8 Further Examples: The Football Players and the Cooperative Cutters 6.9 The Existence of the Perpetuated Dilemma 6.10 Various Problems CHAPTER 7 Afferent or Universal Surfaces 7.1 Introduction 7.2 Universal Surfaces with Null Integrand 7.3 "Universal Surfaces with Linear Vectograms, An Intuitive Purview" 7.4 The Analytic Necessary Condition for a Linear Vectogram Type Universal Service 7.5 The Workable Condition when n = 3 7.6 Why the Name Universal Surface? 7.7 The Calculus of Variations Viewpoint 7.8 All Strategies Optimal 7.9 The Workable Criterion when n = 4 7.10 A Test for a Void and a Further Necessary Condition for a Universal Service 7.11 Test for a Transition Surface 7.12 Further Discussion fo the Basic Nature of Universal Surfaces and Their Relation to the Euler Equation 7.13 Restoration of the Totality of Control Variables 7.14 Semiuniversal Surfaces CHAPTER 8 Games of Kind 8.1 Introduction 8.2 The Barrier Concept 8.3 The Construction of Semipermeable Surfaces 8.4 Termination of Barriers 8.5 Construction of the Barrier 8.6 Some Brief Examples 8.7 Possible Other Species of Barriers 8.8 Fusion of Games of Kind and Degree CHAPTER 9 Examples of Games of Kind 9.1 The Homicidal Chauffeur Game 9.1A Dogfighting a Highly Mobile Target 9.2 The Game of Two Cars 9.3 The Isotropic Rocket 9.4 The Isotropic Rocket: The Envelope Barrier 9.5 Two Remarkably Dissimilar Games in the Same Setting 9.6 Extensions and Applications of the Deadline Game 9.7 Further Games 9.8 Application to Stability and Control CHAPTER 10 Equivocal Surfaces and the Homicidal Chauffeur Game 10.1 Introduction 10.2 The Homicidal Chauffeur: Geometric Solution of the Game of Kind 10.3 The Primary Solution of the Homicidal Chauffeur Game of Degree 10.4 The Universal Curve and Its Tributaries 10.5 Equivocal Surfaces 10.6 An Example with an Equivocal Surface: Preliminaries 10.7 An Example with an Equivocal Surface: Solution 10.8 Discussion of Equivocal Surfaces 10.9 The Equivocal Phenomenon in the Homicidal Chauffeur Game 10.10 Appendix CHAPTER 11 The Application to Warfare 11.1 Game Theory and War 11.2 The Available Techniques 11.3 Types of Applications 11.4 The Broader Problems of Combat 11.5 Problems of Formulation 11.6 The War of Attrition and Attack: A Study 11.7 The Battle of Bunker Hill 11.8 Some Pitfalls in Adapting Game Theory to Warfare 11.9 War of Attrition and Attack: Second Version CHAPTER 12 Toward a Theory with Incomplete Information 12.1 Introduction 12.2 A Speculative Purview 12.3 Search Games with Immobile Hiders 12.4 Search Games with Mobile Hiders 12.5 The Importance of Approximations 12.6 The Chancifying Method APPENDIX A1. A Hit Probability Payoff A2. The Fixed Battery Pursuit Game A3. Optimal Trajectories of Guided Missiles A4. An Illustration from Control Theory A5. The Bomber and Battery Game REFERENCES INDEX
