## Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and OptimizationOne 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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analysis appear apply barrier calculus of variations capture region capture zone Chapter choice circle constant constraint construct control variables coordinates course curvature curve deadline game differential equations differential games direction discrete dispersal surface equivocal surface Euler equation Example fire formal function game of kind game theory homicidal chauffeur game ideas initial conditions instance integral kinematic equations latter Lemma linear vectograms mathematical maximize minimax minimizing missile mixed strategy move obtain one-player game optimal paths optimal play optimal strategies penetration plane player position possible pursuer pursuit game radius realistic space reduced space Research Problem result Section semipermeable surface side similarly simple motion singular surfaces solution solved speed starting points straight suppose tangent target terminal payoff theorem transition surface traversing tributary paths universal surface upper half-plane useable values vector velocity vertical weapons zero

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Page xiii - This quantity is ihepayoffand the game becomes one of degree. It is often possible to find such a continuous payoff in such a way that the aforementioned discrete criteria are automatically subsumed within it. For example, suppose we are interested only in whether capture can be achieved or not. We can pick as payoff the time of capture, with P's objective being to make this quantity as small as possible and £'s, as great.

Page xiii - If the original desideratum was whether or not £ could attain a certain proximity to a certain target, we can make the payoff the distance from the target when capture takes place. By having P strive to maximize this quantity, we are assured that he will not only attain his objective of protecting the target when possible but also the biggest margin of safety, or smallest deficiency if he cannot frustrate £. We answer our question as to what is meant by "best" in all cases by deciding on a numerically...