  This paper is concerned with the closed-loop Stackelberg strategy for
linear-quadratic leader-follower game. Completely different from the open-loop
and feedback Stackelberg strategy, the solvability of the closed-loop solution
even the linear case remains challenging. The main contribution of the paper is
to derive the explicitly linear closed-loop Stackelberg strategy with one-step
memory in terms of Riccati equations. The key technique is to apply the
constrained maximum principle to the leader-follower game and explicitly solve
the corresponding forward and backward difference equations. Numerical examples
verify the effectiveness of the results, which achieves better performance than
feedback strategy.
