A Brief Tutorial On Differential Games

Abstract

A differential game essentially is a game in extensive form, being played in continuous time. For the analysis of the game it has proven to be expedient to look at a normal form representation of the game. It turns out that the latter can be analyzed by techniques of optimal control theory (typically, the maximum principle and Hamilton-Jacobi-Bellman equations of dynamic programming). In a noncooperative game, given the other players’ choice of strategies, each player solves her own individual one-person dynamic decision problem, that is, she solves an optimal control problem. The simultaneous solution of these control problems leads to the characterization of an equilibrium of the game. In cooperative differential games, a Pareto optimal solution can be identified by optimal control methods, by constructing a joint payoff functional for the players.