Differential Games

When there are more than a single decision-maker, each having one’s own objective function that each is trying to maximize, subject to a set of differential equations, then we require an extension of the optimal control theory referred to as the theory of differential games. While representing a generalization of optimal control problems in cases where there is more than one controller or player, differential games are conceptually far more complex than optimal control problems in the sense that it is no longer obvious what constitutes a solution. Indeed, there are different types of solutions such as minimax, Nash, and Stackelberg. We discuss minimax solutions for two-person zero-sum differential games in Sect. 13.1, where one player maximizes his objective function and the other minimizes the same function. Section 13.2 considers nonzero-sum games where all players make simultaneous moves over and each player aims to maximize his own objective function. These are formulated as Nash differential games and their solutions in terms of open-loop and feedback equilibria are discussed. We also apply the theory to a common-property fishery resources game. In Sect. 13.3, we solve a feedback Nash stochastic differential game in advertising. In Sect. 13.4, we discuss a Stackelberg stochastic differential game in which two players make their decisions hierarchically. The player having the right to move first is called the leader and the other player is called the follower. The game is one of cooperative advertising between a manufacturer as the leader deciding on a percentage of the advertising expenditure that he will contribute toward the advertising expenditure of the retailer as the follower. The equilibrium feedback solution that maximizes the objective function of each player is obtained. There are many exercises at the end of the chapter.