In
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. 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
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
problems. In an optimal control problem there is single control
and a single criterion to be optimized; differential game theory generalizes this to two controls
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, indivi ...
, 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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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
Yuliy Sannikov (born November 3, 1978) is a Ukrainian economist known for his contributions to mathematical economics, game theory, and corporate finance. He is an economics professor at the Stanford Graduate School of Business, and won both ...
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
The American Economic Association (AEA) is a learned society in the field of economics. It publishes several peer-reviewed journals acknowledged in business and academia. There are some 23,000 members.
History and Constitution
The AEA was esta ...
'' 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
An autonomous robot is a robot that acts without recourse to human control. The first autonomous robots environment were known as Elmer and Elsie, which were constructed in the late 1940s by W. Grey Walter. They were the first robots in history t ...
.
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.
See also
*
Lotka–Volterra equations
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a pred ...
*
Mean-field game theory Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is insp ...
Notes
Further reading
*
*
External links
*
{{DEFAULTSORT:Differential Game
Control theory
Game theory game classes
Ballistics
Pursuit–evasion