A Three-level Stochastic Linear-quadratic Stackelberg Dierential Game with Asymmetric Information

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A Three-level Stochastic Linear-quadratic

Stackelberg Diﬀerential Game with Asymmetric

Information ∗

Kaixin Kang†, Jingtao Shi‡

October 24, 2022

Abstract: This paper is concerned with a three-level stochastic linear-quadratic Stackelberg

diﬀerential game with asymmetric information, in which three players participate credited as

Player 1, Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player

2 acts as the leader of Player 1 and Player 1 acts as the follower. The asymmetric informa-

tion considered is: the information available to Player 1 is based on the sub-σ-algebra of the

information available to Player 2, and the information available to Player 2 is based on the sub-

σ-algebra of the information available to Player 3. By maximum principle of forward-backward

stochastic diﬀerential equations and optimal ﬁltering, feedback Stackelberg equilibrium of the

game is given with the help of a new system consisting of three Riccati equations.

Keywords: Stackelberg diﬀerential game; linear-quadratic optimal control; maximum principle;

forward-backward stochastic diﬀerential equation; Riccati equation; stochastic ﬁltering

Mathematics Subject Classiﬁcation: 93E20, 60H10, 49K45, 49N70, 91A23

1 Introduction

In this paper, we will study a three-level stochastic linear-quadratic (LQ for short) Stackelberg

diﬀerential game with asymmetric information. The Stackelberg game, also called leader-follower

game, which was proposed by Stackelberg [15] in 1952, is a kind of game with hierarchical

structure. In Stackelberg game, players play the role of the leader or the follower, and make

decision sequentially. First of all, we give an example in the supply chain management to

introduce the research motivation of this paper.

∗This work is supported by National Natural Science Foundations of China (Grant Nos. 11971266, 11831010,

11571205), and Shandong Provincial Natural Science Foundations (Grant Nos. ZR2020ZD24, ZR2019ZD42).

†School of Mathematics, Shandong University, Jinan 250100, P.R. China, E-mail: 202012062@mail.sdu.edu.cn

‡School of Mathematics, Shandong University, Jinan 250100, P.R. China, E-mail: shijingtao@sdu.edu.cn

arXiv:2210.11808v1 [math.OC] 21 Oct 2022

Example 1.1. (Cooperative advertising and pricing problem) He et al. [6] studied a cooper-

ative advertising and pricing problem, in which there are two players, a manufacturer and a

retailer. Chutani and Sethi [5] considered a cooperative advertising problem under manufacturer

and retailer level competition, with a ﬁnite number of independent manufacturers and retailers.

Kennedy et al. [7] extended this problem to the one in a dynamic three-echelon supply chain,

which is composed of a manufacturer, a distributor and a retailer. In their supply chain, the

manufacturer sells his product to the retailer via the distributor.

We consider the following cooperative advertising and pricing model, which is an extension

of that introduced in [7]:











dx(t) = ψR(t)αR(t) + ψD(t)αD(t) + ψM(t)αM(t)p1−x(t)−δx(t)dt

+σ(x(t))dW (t), t ∈[0, T ],

x(0) = x0,

(1.1)

where x(·)is the market awareness which determines the total sales, the constant δ > 0reﬂects

the rate which potential consumers are lost. σ(x)is a variance term, which is usually taken as

σ(x) = Cσpx(1 −x), for some constant Cσ. The retailer decides the retail price PR(·)and sets

the local advertising eﬀort αR(·)with the advertising eﬀectiveness ψR(·). The distributor decides

the distributor price PD(·)and sets the distributor’s advertising eﬀort αD(·)with the advertising

eﬀectiveness ψD(·). The manufacturer decides a wholesale price PM(·), a national advertising

eﬀort αM(·)with the advertising eﬀectiveness ψM(·)and a subsidy rate φ(·)to the retailer’s local

advertising eﬀort through a vertical cooperative advertising program.

Set vR(·),(PR(·), αR(·), ψR(·)),vD(·),(PD(·), αD(·), ψD(·)) and vM(·),(PM(·), αM(·),

ψM(·), φ(·)), whose values are taken from some admissible control sets VR,VDand VMrespec-

tively. Then we encounter a stochastic Stackelberg diﬀerential game with three players. In detail,

ﬁrst the retailer’s optimal strategy v∗

R(·)is solved by:

JR(v∗

R(·), vD(·), vM(·)) = max

vR(·)∈ VR

JR(vR(·), vD(·), vM(·)),∀vD(·), vM(·),(1.2)

with

JR(vR(·), vD(·), vM(·))

=EZT

e−rt(PR(t)−PD(t))D(PR(t))x(t)−(1 −φ(t))α2

R(t)dt.

(1.3)

Then the distributor’s optimal strategy v∗

D(·)can be obtained by:

JD(v∗

R(·), v∗

D(·), vM(·)) = max

vD(·)∈ VD

JD(v∗

R(·), vD(·), vM(·)),∀vM(·),(1.4)

where

JD(vR(·), vD(·), vM(·))

=EZT

e−rt(PD(t)−PM(t))D(PR(t))x(t)−kDD(PR(t))x(t)−α2

D(t)dt.

(1.5)

Finally, the manufacturer’s optimal strategy v∗

Mcould be given by

JM(v∗

R(·), v∗

D(·), v∗

M(·)) = max

vM(·)∈ VM

JM(v∗

R(·), v∗

D(·), vM(·)),(1.6)

with

JM(vR(·), vD(·), vM(·))

=EZT

e−rt(PM(t)−c)D(PR(t))x(t)−kMD(PR(t))x(t)−α2

M(t)−φ(t)α2

R(t)dt.

(1.7)

In the above, r > 0is the discount rate, c > 0is the manufacturing cost, kMand kDare the

transport cost. 0≤D(p)≤1is some demand function satisfying usual conditions.

This is a three-level stochastic Stackelberg diﬀerential game with three players. Each player

hopes to maximize his/her target functional by selecting an appropriate control.

In supply chain management problems, three-level supply chains are often encountered. For

example, 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.

(a) (b)

Figure 1: (a) A schematic of the three-level supply chain; (b)An example of the three-level

supply chain

A schematic of the three-level supply chain is given in Figure 1.(a), and there can be multiple

agents as suppliers, distributors and retails. Another example of the three-level supply chain is

given in Figure 1.(b), in which one agent is the supplier, two agents are the distributors, and

three agents are the retailers. The matrix can indicate whether there is a leadership relationship

between agents, where the leadership relationship means that the information of the “follower”

can be obtained by the “leader”. In Figure 1.(b), for example, S1has a leadership relationship

with D1, then the position of the matrix (1,2) is 1, D1has no leadership relationship with R3,

then the position of the matrix (2,6) is 0. Because the leadership of the supply chain is one-way,

the shaded part of the matrix must be 0. If no distributors exist, it is a two-level stochastic

Stackelberg diﬀerential game.

About stochastic Stackelberg diﬀerential games with one leader and one follower, there are

many related research, such as Yong [22] and applications to newsvendor-manufacturer problem

(Øksendal et al. [11]), principal-agent problem (Williams [19]) and insurer-reinsurer problem

(Chen and Shen [3, 4]), etc. Mukaidani and Xu [10] studied a stochastic Stackelberg diﬀerential

game with one leader and multiple followers. Wang and Zhang [16] studied a stochastic LQ

Stackelberg diﬀerential game of mean-ﬁeld type with one leader and two followers. Wang and

Yan [18] researched a Pareto-based stochastic Stackelberg diﬀerential game with multi-followers.

However, practically in Stackelberg diﬀerential game, due to the emergence of various fac-

tors, players often can not observe the complete information, but can only grasp part of the

information. This kind of problem is called Stackelberg diﬀerential game with asymmetric in-

formation. Shi et al. [12, 13] studied the two-level stochastic Stackelberg diﬀerential games

with asymmetric information, in which the information available to the follower is based on

the sub-σalgebra of that available to the leader. Shi et al. [14] studied a two-level stochastic

LQ Stackelberg diﬀerential game with overlapping information, in which the information of the

follower and the leader has some overlapping parts, but no mutual inclusion relationship. Li et

al. [9] investigated a two-level stochastic LQ Stackelberg diﬀerential game under asymmetric

information patterns, where the follower uses his observation information to design his strategy

whereas the leader implements his strategy using complete information. Zheng and Shi [25, 26]

investigated two-level stochastic Stackelberg diﬀerential games with partial observation, in which

both the leader and the follower have their own observation equations, and the information ﬁl-

tration available to the leader is contained in that to the follower. Yuan et al. [20] discussed a

robust reinsurance contract with asymmetric information in a stochastic Stackelberg diﬀerential

game. Zhao et al. [24] discussed a stochastic LQ Stackelberg diﬀerential game with two leaders

and two followers under an incomplete information structure. See more relevant research in the

monograph by Ba¸sar and Olsder [2], and the review paper by Li and Sethi [8].

Motivated by the above three-level supply chain and the related literatures about the stochas-

tic Stackelberg diﬀerential game, in this paper we study a three-level stochastic LQ Stackelberg

diﬀerential game with asymmetric information. We call the players in the game as Player 1,

Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player 2 acts as

the leader of Player 1 and Player 1 acts as the follower. The asymmetric information consid-

ered is: the information available to Player 1 is based on the sub-σ-algebra of the information

available to Player 2, and the information available to Player 2 is based on the sub-σ-algebra

of the information available to Player 3. By maximum principle and optimal ﬁltering, feedback

Stackelberg equilibrium of the game is given with the help of a new system consisting of three

Riccati equations.

The rest of this paper is organized as follows. In Section 2, we formulate our problem.

Section 3 is devoted to ﬁnd the feedback Stackelberg equilibrium of the game. Finally in Section

4, some concluding remarks are given.

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AThree-levelStochasticLinear-quadraticStackelbergDierentialGamewithAsymmetricInformation*KaixinKang,JingtaoShiOctober24,2022Abstract:Thispaperisconcernedwithathree-levelstochasticlinear-quadraticStackelbergdierentialgamewithasymmetricinformation,inwhichthreeplayersparticipatecreditedasPlayer1,Pl...

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A Three-level Stochastic Linear-quadratic Stackelberg Dierential Game with Asymmetric Information

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