Stackelberg equilibria are a delicate matter in game theory in general and in differential game theory in particular, where the critical aspects of hierarchical play become evident in terms of the time inconsistency issue that usually affects these structures. Consequently, this chapter is not a recollection of Stackelberg games in IO, but rather a reconstruction of what differential game theory can say about the solution of dynamic Stackelberg models, complemented by the exposition of a few applications in the field covered by this volume. In a sense, the following treatment of this subject can be taken as an invitation to intensify the implementation of Markovian Stackelberg solutions along a number of different directions where this approach could yield relevant results but its technical difficulties have long prevented its adoption.

As stated in Chapter 1 (Section 1.5), differential game theory becomes aware of the time inconsistency affecting open-loop Stackelberg solutions from Simaan and Cruz (1973a,b) and Kydland (1977) onwards, and quite quickly relevant applications appear in the literature on economic policy (Kydland and Prescott, 1977; Calvo, 1978; Barro and Gordon, 1983a,b), in a debate focussing upon rules versus discretion in manoeuvring monetary and fiscal policy instruments.

A very important aspect which I would like to stress most explicitly is that the presence of time inconsistency affecting an open-loop Stackelberg outcome (note that I'm avoiding the use of the term ‘equilibrium’) implies that we cannot expect to observe the realization of that outcome, except at most for an instant (in the model) or a very restricted time interval (in the real world) because players will deviate from it either immediately (again, in the theoretical model) or very quickly (again, in real world situations).

In the context of games dealing with monetary and fiscal policies, this problem clearly emerges in Cohen and Michel (1988, pp. 267–68):

The nature of the problem can be easily grasped from the original static formulation of the sequential play Cournot game in Stackelberg (1934). The subject matter of the game can be spelled out in the following terms.