In 1971, Avner Friedman published Differential Games, a book which developed a mathematical framework for studying games in which the players' positions evolve continuously over time. These games are of particular interest to engineers and economists, although the mathematical theory underlying these games will also be of independent interest to mathematicians who enjoy analysis and differential equations. Friedman's book has recently been republished by Dover in an unabridged version geared towards graduate students. Friedman assumes knowledge of advanced calculus and Lebesgue integration, and, while he introduces most of the necessary results in differential equations, he does so at a breakneck speed and it's hard to imagine anyone learning everything they need from this introduction.
Most of Friedman's book deals with zero-sum games with two opposing players, which fall into different categories such as pursuit games (the original differential games introduced by Rufus Isaacs in 1951), games of survival, games of fixed duration, and games of delayed information. These names sound like something you might find on a reality TV show, but for the most part Friedman sticks with the mathematical theory rather than any applications or even explanations of the motivation or terminology. His emphasis is on proofs for the existence of solutions and saddle points and the connection between these solutions and the solutions to various systems of differential equations. In the final chapter, Friedman considers the case of games with more than two players and he proves the existence of solutions to these games under various conditions.
Darren Glass is an assistant professor at Gettysburg College whose mathematical interests include number theory, Galois theory, and cryptography. He can be reached at firstname.lastname@example.org
|1.||Definition of a Differential Game|
|2.||Games of Fixed Duration|
|3.||Games of Pursuit and Evasion|
|4.||Computation of Saddle Points|
|5.||Games of Survival|
|6.||Games with Restricted Phase Coordinates|
|Index of Conditions|