Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization

CHAPTER 1 An Introduction

The Theory of Games, 1
The State and Control Variables, 3
Battle Games, 4
Games with Moving Craft, 5
Pursuit Games, 8
Games of Kind and Games of Degree, 13 Strategies, 14
Dogfights, Firing Games, Programming, and Athletics, 15
Two Examples, 16
A Perspective on Precision, 20
A Perspective on Progress, 22
On Reading This Book, 24

CHAPTER 2 Definitions, Formulation, and Assumptions
2.1 The Kinematic Situation, 25
2.2 The Realistic and Reduced Space, 28
2.3 Termination of the Game, 30
2.4 The Payoff, 32
2.5 Games of Kind and Games of Degree, 35
2.6 Strategies, 36
2.7 Canonization of the Vectograms, 40
2.8 A Lemma on Circular Vectograms, 43

CHAPTER 3 Discrete Differential Games 44
3.1 Introduction, 44
3.2 The General Discrete Game, 45
3.3 Battles of Extinction, 50
3.4 Two Discrete Pursuit Games, 56
3.5 Quasi-Discrete Games, 62

CHAPTER 4 The Basic Mathematics and the Solution
Technique in the Small 64
4.1
The Nature of a Solution, 64
The Main Equation, 67
Semipermeable Surfaces and a Second Deri- vation of the Main Equation, 70
The Verification Theorem, 72
The Path Equations, 80
The Retrogression Principle, 81
The Initial Conditions, 82

CHAPTER 5 Mainly Examples: Transition Surfaces:
Integral Constraints 86
5.1 Transition Surfaces, 87
5.2 The Dolichobrachistochrone, 88
5.3 The Relationship to the Euler Equation, 95
5.4 The War of Attrition and Attack, 96
5.5 The Isotropic Rocket Pursuit Game, 105
5.6 An Optimal Program of Steel Production, 116
5.7 Integral Constraints, 124

CHAPTER 6 Efferent of Dispersal Surfaces 132
Singular Surfaces, 132
Dispersal Surfaces, 134
The Nature of Dispersal Surfaces, 135
The Question of the Perpetuated Dilemma, 137 The Construction of Dispersal Surfaces, 140 Further Examples, 142
The Geometric Method for Simple Pursuit Games of Kind, 144

 CHAPTER 7
CONTENTS XV
6.8 Further Examples: The Football Players and the Cooperative Cutters, 145
6.9 The Existence of the Perpetuated Dilemma, 149 6.10 Various Problems, 152
Afferent or Universal Surfaces 156
7.1 Introduction, 156
7.2 Universal Surfaces with Null Integrand, 157
7.3 Universal Surfaces with Linear Vectograms,
An Intuitive Purview, 161
7.4 The Analytic Necessary Condition for a Linear
Vectogram Type Universal Service, 166
7.5 The Workable Condition when n = 3, 168
7.6 Why the Name Universal Surface? 173
7.7 The Calculus of Variations Viewpoint, 173
7.8 All Strategies Optimal, 175
7.9 The Workable Criterion when n = 4, 177
7.10 A Test for a Void and a Further Necessary
Condition for a Universal Service, 187
7.11 Test for a Transition Surface, 188
7.12 Further Discussion of the Basic Nature of
Universal Surfaces and Their Relation to the
Euler Equation, 189
7.13 Restoration of the Totality of Control Vari-
ables, 190
7.14 Semiuniversal Surfaces, 196
Games of Kind 200
8.1 Introduction, 201
8.2 The Barrier Concept, 202
8.3 The Construction of Semipermeable Sur-
faces, 205
8.4 Termination of Barriers, 210
8.5 Construction of the Barrier, 214
8.6 Some Brief Examples, 223
8.7 Possible Other Species of Barriers, 228
8.8 Fusion of Games of Kind and Degree, 230
CHAPTER 8
CHAPTER 9 Examples of Games of Kind 231
9.1 The Homicidal Chauffeur Game, 232 9.lA. Dogfighting a Highly Mobile Target, 237 9.2 The Game of Two Cars, 237

 xvi CONTENTS
9.3 The Isotropic Rocket, 244
9.4 The Isotropic Rocket: The Envelope
Barrier, 250
9.5 Two Remarkably Dissimilar Games m the
Same Setting, 255
9.6 Extensions and Applications of the Deadline
Game, 265
9.7 Further Games, 270
9.8 Application to Stability and Control, 272
CHAPTER 10 Equivocal Surfaces and the Homicidal Chauffeur Game
273
10.1 10.2
10.3
10.4 10.5 10.6
10.7
10.8 10.9
Introduction, 273
The Homicidal Chauffeur: Geometric Solution of the Game of Kind, 273
The Primary Solution of the Homicidal Chauffeur Game of Degree, 278
The Universal Curve and Its Tributaries, 280 Equivocal Surfaces, 282
An Example
Preliminaries,
An Example
Solution, 290
Discussion of
The Equivocal Phenomenon in the Homicidal Chauffeur Game, 297
with an Equivocal Surface: 286
with an Equivocal Surface: Equivocal Surfaces, 295
10.10 Appendix, 301
CHAPTER 11 The Application to Warfare 305
11.1 Game Theory and War, 305
11.2 The Available Techniques, 306
11.3 Types of Applications, 308
11.4 The Broader Problems of Combat, 309
11.5 Problems of Formulation, 311
11.6 The War of Attrition and Attack: A
Study, 312
11.7 The Battle of Bunker Hill, 316
11.8 Some Pitfalls in Adapting Game Theory to
Warfare, 329
11.9 War of Attrition and Attack: Second
Version, 330

 CONTENTS xvii
CHAPTER 12 Toward a Theory with Incomplete Information 336
12.1 Introduction, 336
12.2 A Speculative Purview, 337
12.3 Search Games with Immobile Hiders, 345
12.4 Search Games with Mobile Hiders, 349
12.5 The Importance of Approximations, 350
12.6 The Chancifying Method, 354
APPENDIX
Al. A Hit Probability Payoff, 357
A2. The Fixed Battery Pursuit Game, 358
A3. Optimal Trajectories of Guided Missiles,361
A4. An Illustration from Control Theory, 366
A5. The Bomber and Battery Game, 368
REFERENCES 377 INDEX 379