This article considers the hierarchical decision problems with $H_\infty$ constraint, where the proposed feedback Stackelberg strategy incorporates a two-level control progress. The leader first announces his action at the beginning of the game and anticipates the follower's optimal response; the follower chooses a response to optimize his/her cost function with the information of the leader's action. The optimal action of the leader, coupled with the rational response of the follower, forms a Stackelberg equilibrium. Both the leader and the follower exchange their observations and historical control inputs. Under the assumption of linear feedback strategies, the problems are converted into unsolved feedback gain matrix. The introduced Pontryagin's maximum principle develops a solution based on the forward and backward stochastic difference equations. The separation principle is proposed to deal with coupled state estimation gains and optimal controller gains. To this end, the necessary and sufficient solvability conditions for both finite horizon and infinite horizon case are derived.