On coincidence of feedback and global Stackelberg equilibria in a class of differential games

Abstract

This paper shows for a class of differential games that the global Stackelberg equilibrium (GSE) coincides with the feedback Stackelberg equilibrium (FSE), although the GSE assumes that the leader/regulator an- nounces at the initial time the regulatory instrument rule she will follow for the rest of the game, while in the FSE, the regulator at any time chooses the optimal level of the regulatory instrument rate. This coincidence is based on the fact that the FSE is calculated using dynamic programming what implies that although the regulator chooses the regulatory instrument rate level that maximizes social welfare, the first-order condition for the maximization of the right-hand side of the Hamilton-Jacobi-Bellman equa- tion implicitly defines a rule for the regulatory instrument. Then, as the regulatory instrument rule de- fined by the FSE implements the efficient outcome as the GSE does, the rules defined by both equilibria must be the same. In the second part of the paper, we check that this is the case for two examples. The first is an operations research model, while the second is an economic model. The first example fits in a linear-state differential game structure, while the second example presents a linear-quadratic specifica- tion. In both cases the regulatory instrument rules for both equilibria (GSE and FSE) are calculated and identical expressions are obtained.