Optimal Stackelberg-Nash Strategy for One Leader and Two Followers

In this work, we study the Stackelberg-Nash game for one leader and two followers. The open-loop solution is explicitly provided in terms of several decoupled and symmetric Riccati equations. Then our results have less computation than most of the existing results, as they need to solve some coupled and nonsymmetric Riccati equations. The main difficult is how to establish the nonhomogeneous relation between the forward and backward variables obtained in the problem, and we solved this problem by introducing some new costates.