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

by Rufus Isaacs

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.

Product Details

ISBN-13:	9780486135984
Publisher:	Dover Publications
Publication date:	03/29/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	416
File size:	23 MB
Note:	This product may take a few minutes to download.

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

Editorial Reviews

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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