Game theory is an extension of control theory that deals with situations where several players exercise authority over a system. In general, each player may pursue its own objective. The players are not necessarily adversaries; they may choose to cooperate if it is to their advantage. In addition, the players may not know everything there is to know. They may act to learn what they do not know, or exploit the situation based on what they do know. This is a typical exploration-exploitation trade-off.

Game theory has applications to economics (several Nobel Prizes were awarded in this field), international diplomacy, guidance and pursuit evasion, warfare, and sports. Game theory is also applicable to control (see Figure 1.4), where the exogenous and endogenous inputs are two players in a game.

Section 10.1 presents a taxonomy of two-player games. Section 10.2 describes an example of a simple game of pursuit evasion in a two-player football scrimmage. Section 10.3 describes the Bellman-Isaacs equation. Sections 10.4 and 10.5 present modeling and features of the solution for the homicidal chauffeur game. Section 10.6 describes a game-theoretic view of proportional navigation. Sections 10.7, 10.8, and 10.9 present a summary of the key results in the chapter, bibliographic notes for further reading, and homework problems, respectively.

Taxonomy of Two-Player Games

We consider two-player games, in which the two players have control actions u and v, respectively.