In game theory, differential games are a group of problems related to the modeling and analysis of conflict in the context of a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

. More specifically, a

state variable A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...

or variables evolve over time according to a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

. Early analyses reflected military interests, considering two actors—the pursuer and the evader—with diametrically opposed goals. More recent analyses have reflected engineering or economic considerations.

Connection to optimal control

Differential games are related closely with optimal control problems. In an optimal control problem there is single control

u(t)

and a single criterion to be optimized; differential game theory generalizes this to two controls

u_(t),u_(t)

and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve its goal; the system responds to the inputs of all players.

History

In the study of

competition Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, indiv ...

, differential games have been employed since a 1925 article by Charles F. Roos. The first to study the formal theory of differential games was Rufus Isaacs, publishing a text-book treatment in 1965. One of the first games analyzed was the 'homicidal chauffeur game'.

Random time horizon

Games with a random time horizon are a particular case of differential games. In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectancy of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval

Applications

Differential games have been applied to economics. Recent developments include adding

stochasticity Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...

to differential games and the derivation of the stochastic feedback Nash equilibrium (SFNE). A recent example is the stochastic differential game of capitalism by Leong and Huang (2010). In 2016 Yuliy Sannikov received the

John Bates Clark Medal The John Bates Clark Medal is awarded by the American Economic Association to "that American economist under the age of forty who is adjudged to have made a significant contribution to economic thought and knowledge." The award is named after the ...

from the '' American Economic Association'' for his contributions to the analysis of continuous-time dynamic games using

stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

methods. Additionally, differential games have applications in

missile guidance Missile guidance refers to a variety of methods of guiding a missile or a guided bomb to its intended target. The missile's target accuracy is a critical factor for its effectiveness. Guidance systems improve missile accuracy by improving its P ...

and autonomous systems. For a survey of

pursuit–evasion Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early ...

differential games see Pachter.

Notes

External links

* {{DEFAULTSORT:Differential Game Control theory Game theory game classes Ballistics Pursuit–evasion