A Three-level Stochastic Linear-quadratic Stackelberg Differential Game

The Stackelberg game describes the game problem when the player’s position is asymmetric,the players of the Stackelberg game make decisions at different times.Stackelberg game theory has a wide range of applications in finance,engineering,management and other fields.In the classic Stackelberg game,there are usually only two players involved,namely the leader and the follower.For a multinational company with multiple sales markets,it is difficult for suppliers to adjust their behavior in direct response to retailers,and the presence of distributors is necessary.This forms a three-level supply chain of suppliers,distributors and retailers.Based on the background of this three-level supply chain problem,this paper studies the equilibrium strategy of three-level stochastic linear quadratic Stackelberg differential game,and studies it from the aspects of information symmetry and information asymmetry.In the third chapter,this paper studies the state feedback representation of the equilibrium strategy of the three-level stochastic linear quadratic Stackelberg game with asymmetric information.There are three players involved in this model,namely Player 1,Player 2,and Player 3,which Player 3 acts as the leader of Player 2 and Player 1;Player 2 acts as the leader of Player 1,but also as a follower of Player 3;Player 1 acts as a follower of Player 2 and Player 3.The information asymmetry considered is:the information available to Player 1 is a sub-σ-algebra of the information available to Player 2,and the information available to Player 2 is a sub-σ-algebra of the information available to Player 3.To solve the equilibrium strategy,we solve the optimal control problems of Player 1,Player 2 and Player 3 in turn.Under such asymmetric information,for the optimal control problem of Player 1 and Player 2,the optimal strategy can be solved by the stochastic maximum principle of partial information and the verification theorem.The state feedback representation of the player’s equilibrium strategy is obtained by introducing the Riccati equations.At this time,the state equation for the optimal problem of Player 3 is a coupled forward-backward stochastic differential equation,and the state equation contains two kinds of random filtering.This paper uses a direct construction method to get the equilibrium strategy of Player 3.A set of high-dimensional Riccati equations is introduced to get state feedback representation of the Player 3’s equilibrium strategy.In the fourth chapter,this paper studies the state feedback representation of the equilibrium strategy of three-level stochastic linear quadratic Stackelberg differential game with symmetric information.The leadership situation between the three players involved is the same as in the third chapter.Using the theorems in Chapter 3,the player’s equilibrium strategy can be solved.When expressing the equilibrium strategy as a state feedback form,the same method as in Chapter 3 is used to obtain the state feedback representation of the player’s equilibrium strategy by introducing high-dimensional Riccati equations.At last,applying this model to a three-level supply chain sales problem consisting of suppliers,distributors and retailers,it is studied how suppliers,distributors and retailers can set their own level of effort so that they can get the maximum benefit.The three-level supply chain and the two-level supply chain are compared and the difference is analyzed.