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I am most familiar with this excellent reference [inside viewable on Amazon]:

Dynamic Noncooperative Game Theory (Classics in Applied Mathematics) (Paperback) by Tamer Basar, Geert Jan Olsder

Question 1 - Are there other recommended references?

The ArXiv groups Game theory with Computer programming.

http://arxiv.org/list/cs.GT/06

Many of the physics examples appear to involve the category of ‘Prisoner's dilemma’ games although Basar and Olsder devote a chapter to the category ‘Pursuit-evasion’ games. From my perspective, the latter appears to be more useful in physics.

Question 2 - What other game categories are used in physics?

Wikipedia [2nd paragraph] lists some areas of game theory applications:

“The field came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies. Game theory is now used in many diverse academic fields, ranging from biology and psychology to sociology and philosophy. Beginning in the 1970s, game theory has been applied to animal behavior, including species' development by natural selection. Because of games like the prisoner's dilemma, in which rational self-interest hurts everyone, game theory has been used in political science, ethics and philosophy. Finally, game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.”

http://en.wikipedia.org/wiki/Game_theory

Question 3 - Could Archimedes use of minimal outer polygons and maximal inner polygons to approximate the circumference of a circle be an early example of game theory application?

Question 4 - Is Game theory powerful enough to unify all disciplines of mathematics?