Differential games with hierarchical play

The preceding chapter dealt with differential games in which all players make their moves simultaneously. We now turn to a class of differential games in which some players have priority of moves over other players. To simplify matters, we focus mostly on the case where there are only two players. The player who has the right to move first is called the leader and the other player is called the follower. A well-known example of this type of hierarchical-moves games is the Stackelberg model of duopoly, which is often contrasted with the Cournot model of duopoly. The plan of this chapter is as follows. In section 5.1 we review the one-shot Cournot duopoly game and the corresponding one-shot Stackelberg game. We also present a modified version of the one-shot Stackelberg game as a quick means of raising the issue of time inconsistency (sometimes referred to as dynamic inconsistency) in Stackelberg games, which we further expound in the rest of the chapter. In sections 5.2 and 5.3 we define the concepts of open-loop Stackelberg equilibrium and (nondegenerate) Markovian Stackelberg equilibrium for differential games. We show in section 5.2 that open-loop Stackelberg equilibria are, in general, not time consistent. There are, of course, exceptions to this rule which we also consider. In section 5.3 we turn to the analysis of nondegenerate Markovian Stackelberg equilibria. In general, it is difficult to find such equilibria. However, we are able to provide some rules of thumb which work in a number of situations.