Editing Game theory (section)

===Differential games===
{{main|Differential game}}
Differential games such as the continuous [[Pursuit-evasion|pursuit and evasion game]] are continuous games where the evolution of the players' state variables is governed by [[differential equation]]s. The problem of finding an optimal strategy in a differential game is closely related to the [[optimal control]] theory. In particular, there are two types of strategies: the open-loop strategies are found using the [[Pontryagin's Minimum Principle|Pontryagin maximum principle]] while the closed-loop strategies are found using [[Hamilton–Jacobi–Bellman equation|Bellman's Dynamic Programming]] method.

A particular case of differential games are the games with a random [[time horizon]].<ref name="PM66">{{cite journal |language=ru |last1=Petrosjan |first1=L. A. |last2=Murzov |first2=N. V. |date=1966 |title=Game-theoretic problems of mechanics |journal=Litovsk. Mat. Sb. |volume=6 |pages=423–433}}</ref> In such games, the terminal time is a random variable with a given [[probability distribution]] function. Therefore, the players maximize the [[mathematical expectation]] 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.