A Mean-Field Stochastic Linear-Quadratic Stackelberg Differetial Game With One Leader And Two Followers

Leader-follower(also named Stackelberg)game is introduced by Stackel-berg’s monograph Market economy theory,in the game,the leader has power of dominations over the follower.Leader-follower game problem has wide applica-tion in lvarious fields,such as physics,economics,finance.Most of the literatures only study leader-follower game with one leader and one follower.In fact,it is important to consider the case of one leader and multiple followers from the point of view of practical application.For example,in the supply chain in-ventory replenishment model with one supplier versus multiple purchasers,the supplier determines the minimum replenishment period of the supplied goods,with corresponding discounts,to motivate the purchaser to arrange purchas-es.Mean-field theory has wide application in various fields,such as physics,economy and finance,engineering,which attracts a lot of scholars engaged in the research of the mean-field theory.Lasry and Lions[1]applied stochastic d-ifferential game and mean-field theory to the financial and economic fields,and studied the Nash equilibrium and N-players differential game problem.Howev-er,little literature has been found on the research in mean-field leader-follower game.Due to this,this paper investigates a stochastic Stackelberg differential game of mean-field type,with one leader and multiple followers.The main feature of leader-follower differential game is that the game in-volves leaders and followers.Without loss of generality,this paper investigates a stochastic Stackelberg differential game with one leader and two followers.The main works are as follows:(1)Suppose the followers are equal and independent,and the leader and the followers possess symmetric information.The open-loop Stackelberg solution is obtained by stochastic maximum principle and verification theorem,meanwhile three new systems of Riccati equations are obtained.If each system of equations has a unique solution,the optimal control is expressed as a feedback form of the state and its mean.(2)Suppose the players possess asymmetric information.The leader’s state equation contains mathematical expectation of state and state filtering,so it is difficult to solve the leader’s problem.A direct construction method is used to solve the problem,and then optimal strategy is given by conjecturing.