Ingame theory, differential games are a group of problems related to the modeling and analysis of conflict in the context of a dynamical system. More specifically, a state variable or variables evolve over time according to a differential equation. 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.[1][2]
Differential games are related closely with optimal control problems. In an optimal control problem there is single control 
In the study of competition, differential games have been employed since a 1925 article by Charles F. Roos.[4] The first to study the formal theory of differential games was Rufus Isaacs, publishing a text-book treatment in 1965.[5] One of the first games analyzed was the 'homicidal chauffeur game'.
Games with a random time horizon are a particular case of differential games.[6] 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[7][8]
Differential games have been applied to economics. Recent developments include adding stochasticity 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).[9] In 2016 Yuliy Sannikov received the John Bates Clark Medal from the American Economic Association for his contributions to the analysis of continuous-time dynamic games using stochastic calculus methods.[10][11]
Additionally, differential games have applications in missile guidance[12][13] and autonomous systems.[14]
For a survey of pursuit–evasion differential games see Pachter.[15]
