A crash course in differential games and applications

The methodology of differential games is a combination of optimal control theory and game theory. It is the natural framework for economic analysis with strategic interaction and dynamical optimization. The theory gained traction by seminal papers in the early seventies, and it gradually found its way into economics. The purpose of this paper is to make theory and applications of differential games easily accessible by explaining the basics and by developing some characteristic applications. The core of the theory focuses on the open-loop and the multiple Markov-perfect Nash equilibria that use the maximum principle and dynamic programming as the techniques to solve the optimal control problems. The applications are the game of international pollution control and the game of managing a lake, which is an example of an ecological system with tipping points. Finally, it is interesting to note that the discovery of time-inconsistency in the open-loop Stackelberg equilibrium had a huge impact on macroeconomics, since policy making under rational expectations is a Stackelberg differential game.