Feedback strategies for discrete-time linear-quadratic two-player descriptor games

Abstract

International audienceThis paper deals with discrete-time linear-quadratic descriptor two-player games. The two frameworks of Nash and Stackelberg strategies with feedback information structure are considered. The aim is to provide sufficient conditions ensuring, for any initial state, the existence and the uniqueness of a tri-trajectory, gathering the state and the controls of the two players, which reaches the equilibrium. The provided results are mainly based on a discrete-time matrix block formulation. The provided algorithms consist in an iterative backward in time procedure, issued from Dynamic Programming. Numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

Necessary/sufficient conditions for Pareto optimum in cooperative difference game

“…() Li et al obtained the team‐optimal Stackelberg strategy with the state feedback information structure for a class of discrete‐time linear‐quadratic (LQ) 2‐person nonzero‐sum dynamic games. Jungers studied the discrete‐time LQ descriptor 2‐player game and provided the sufficient conditions for the existence and uniqueness of the feedback Nash and Stackelberg strategies, respectively. Xu et al researched the Stackelberg strategy for a 2‐player game in the discrete‐time dynamic setting and presented a necessary and sufficient condition in terms of 3 symmetric and decoupled Riccati equations.…”

“…In the past few decades, there have been a lot of works on various strategies of differential game, such as Pareto strategy, 4-8 Stackelberg strategy, [9][10][11] Compared with the differential game, the difference game is far less developed. [16][17][18][19][20] Li et al 17 obtained the team-optimal Stackelberg strategy with the state feedback information structure for a class of discrete-time linear-quadratic (LQ) 2-person nonzero-sum dynamic games. Jungers 16 studied the discrete-time LQ descriptor 2-player game and provided the sufficient conditions for the existence and uniqueness of the feedback Nash and Stackelberg strategies, respectively.…”

“…Plenty of works have been done on the noncooperative game. [12][13][14][15][16]19 In contrast to the noncooperative game, the development of the cooperative game is very slow. The Pareto strategy plays a central role in analyzing cooperative game.…”