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

Overview

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 ...

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

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Overview

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.

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Editorial Reviews

Booknews
A reprint of a 1965 volume which looks at game theory methods for obtaining solutions in conflict situations. Combining principles of game theory, the calculus of variation, and control theory, the author considers an array of problems from military situations and pursuit and evasion tactics to athletic contests and games of firing and maneuver. Annotation c. Book News, Inc., Portland, OR booknews.com
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Product Details

  • ISBN-13: 9780486406824
  • Publisher: Dover Publications
  • Publication date: 1/20/1999
  • Series: Dover Books on Mathematics Series
  • Edition description: Unabridged
  • Pages: 416
  • Sales rank: 1,058,139
  • Product dimensions: 5.44 (w) x 8.51 (h) x 0.82 (d)

Table of Contents

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
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