Feedback Stackelberg Strategy for Discrete-Time Descriptor Games
Résumé
The Stackelberg strategies are suitable for two-person dynamic games, where there exists a hierarchy between the players, respectively called the leader and the follower. When the closed-loop information structure is considered, obtaining the Stackelberg strategies is a difficult issue, due to their time-inconsistency. Nevertheless, when the available information about the trajectory is memoryless, that is restricted to the current state, the Stackelberg strategies, thus called feedback, become strongly time consistent. This property allows to apply the Bellman's principle to the game. In this contribution we will provide a solution via a step by step backward in time numerical algorithm for discrete-time linear quadratic games. This original method is based on a matrix block formulation and the symmetry properties of an Hamiltonian matrix to avoid the canonical singular matrix decomposition. For an infinite time horizon, we will show that an implicit formulation of a Algebraic Riccati type Equation is obtained. Moreover an extension to descriptor games will be provided by using the same approach. Numerical examples illustrate the efficiency of the proposed method.