## 13.5.2 Differential Game Theory

The extension of sequential game theory to the continuous-time case is called differential game theory (or dynamic game theory [59]), a subject introduced by Isaacs [477]. All of the variants considered in Sections 9.3, 9.4, 10.5 are possible:

1. There may be any number of players.
2. The game may be zero-sum or nonzero-sum.
3. The state may or may not be known. If the state is unknown, then interesting I-spaces arise, similar to those of Section 11.7.
4. Nature can interfere with the game.
5. Different equilibrium concepts, such as saddle points and Nash equilibria, can be defined.
See [59] for a thorough overview of differential games. Two players, and , can be engaged in a differential game in which each has a continuous set of actions. Let and denote the action spaces of and , respectively. A state transition equation can be defined as

 (13.203)

in which is the state, , and .

Linear differential games are an important family of games because many techniques from optimal control theory can be extended to solve them [59].

Example 13..17 (Linear Differential Games)   The linear system model (13.37) can be extended to incorporate two players. Let be a phase space. Let and be an action spaces for . A linear differential game is expressed as

 (13.204)

in which , , and are constant, real-valued matrices of dimensions , , and , respectively. The particular solution to such games depends on the cost functional and desired equilibrium concept. For the case of a quadratic cost, closed-form solutions exist. These extend techniques that are developed for linear systems with one decision maker; see Section 15.2.2 and [59].

The original work of Isaacs [477] contains many interesting examples of pursuit-evasion differential games. One of the most famous is described next.

Example 13..18 (Homicidal Chauffeur)   In the homicidal chauffeur game, the pursuer is a Dubins car and the evader is a point robot that can translate in any direction. Both exist in the same world, . The speeds of the car and robot are and , respectively. It is assumed that , which means that the pursuer moves faster than the evader. The transition equation is given by extending (13.15) to include two state variables that account for the robot position:

 (13.205)

The state space is is , and the action spaces are and .

The task is to determine whether the pursuer can come within some prescribed distance of the evader:

 (13.206)

If this occurs, then the pursuer wins; otherwise, the evader wins. The solution depends on the , , , , and the initial state. Even though the pursuer moves faster, the evader may escape because it does not have a limited turning radius. For given values of , , , and , the state space can be partitioned into two regions that correspond to whether the pursuer or evader wins [59,477]. To gain some intuition about how this partition may appear, imagine the motions that a bullfighter must make to avoid a fast, charging bull (yes, bulls behave very much like a fast Dubins car when provoked).

Another interesting pursuit-evasion game arises in the case of one car attempting to intercept another [694].

Example 13..19 (A Game of Two Cars)   Imagine that there are two simple cars that move in the same world, . Each has a transition equation given by (13.15). The state transition equation for the game is

 (13.207)

The pursuit-evasion game becomes very interesting if both players are restricted to be Dubins cars.

Steven M LaValle 2012-04-20