## Abstract

In this chapter, we discuss a leader-follower (also called Stackelberg) stochastic differential game with asymmetric information. Here the word “asymmetric” means that the available information of the follower is some sub- σ -algebra of that available to the leader, though they play as different roles in the classical literatures. Stackelberg equilibrium is represented by the stochastic versions of Pontryagin’s maximum principle and verification theorem with partial information. A linear-quadratic (LQ) leader-follower stochastic differential game with asymmetric information is studied as applications. If some system of Riccati equations is solvable, the Stackelberg equilibrium admits a state feedback representation.

### Keywords

- backward stochastic differential equation (BSDE)
- leader-follower stochastic differential game
- asymmetric information
- stochastic filtering
- linear-quadratic control
- Stackelberg equilibrium

## 1. Introduction

Throughout this chapter, we denote by

### 1.1. Motivation

In practice, there are many problems which motivate us to study the leader-follower stochastic differential games with asymmetric information. Here we present two examples.

*Example 1.1:* (Continuous time principal-agent problem) The principal contracts with the agent to manage a production process, whose cumulative proceeds (or output)

where

where

Thus, the agent earns the same rate of return

We consider an optimal implementable contract problem in the so-called “hidden savings” information structure (Williams [1], also in Williams [2]). In this problem, the principal can observe his asset

is maximized. Here

Let

where

where

*Example 1.2:* (Continuous time manufacturer-newsvendor problem) Let

where

When the manufacturer has a fixed production cost per unit

In the above, we assume that

Let

and then to select a

formulates a leader-follower stochastic differential game with asymmetric information. In this setting, the manufacturer is the leader and the retailer is the follower. Any process triple

### 1.2. Problem formulation

Motivated by the examples earlier, in this chapter we study the leader-follower stochastic differential games with asymmetric information. Let *stochastic differential equation* (SDE)

where

Let us now explain the asymmetric information character between the follower (player 1) and the leader (player 2) in this chapter. Player 1 is the follower, and the information available to him at time

The game initiates with the announcement of the leaders control

Here

**SOCPF.** For any chosen

subject to Eqs. (13) and (15). Such a

In the following step, once knowing that the follower will take such an optimal control

Here

**SOCPL.** Find a

subject to Eqs. (13) and (17). Such a *leader-follower stochastic differential game with asymmetric information*. If there exists a control process pair *Stackelberg equilibrium*.

In this chapter, we impose the following assumptions.

**(A1.1)** *For each* *, the functions* *are twice continuously differentiable in* *. For each* *, functions* *and* *are continuously differentiable in* *and* *, respectively. Moreover, for each* *and any* *, there exists* *such that*

for

### 1.3. Literature review and contributions of this chapter

Differential games are initiated by Issacs [4], which are powerful in modeling dynamic systems where more than one decision-makers are involved. Differential games have been researched by many scholars and have been applied in biology, economics, and finance. Stochastic differential games are differential games for stochastic systems involving noise terms. See Basar and Olsder [5] for more information about differential games. Recent developments for stochastic differential games can be seen in Hamadène [6], Wu [7], An and Øksendal [8], Wang and Yu [9, 10], and the references therein.

Leader-follower stochastic differential game is the stochastic and dynamic formulation of the Stackelberg game, which was introduced by Stackelberg [11] in 1934, when the concept of a hierarchical solution for markets where some firms have power of domination over others, is defined. This solution concept is now known as the Stackelberg equilibrium, which in the context of two-person nonzero-sum games, involves players with asymmetric roles, one leader and one follower. Pioneer study for stochastic Stackelberg differential games can be seen in Basar [12]. Specifically, a leader-follower stochastic differential game begins with the follower aims at minimizing his cost functional in response to the leader’s decision on the whole duration of the game. Anticipating the follower’s optimal decision depending on his entire strategy, the leader selects an optimal strategy in advance to minimize his cost functional, based on the stochastic Hamiltonian system satisfied by the follower’s optimal decision. The pair of the leader’s optimal strategy and the follower’s optimal response is known as the Stackelberg equilibrium.

A linear-quadratic (LQ) leader-follower stochastic differential game was studied by Yong [13] in 2002. The coefficients of the the cost functionals and system are random, the diffusion term of the state equation contain the controls, and the weight matrices for the controls in the cost functionals are not necessarily positive definite. The related Riccati equations are derived to give a state feedback representation of the Stackelberg equilibrium in a nonanticipating way. Bensoussan et al. [14] obtained the global maximum principles for both open-loop and closed-loop stochastic Stackelberg differential games, whereas the diffusion term does not contain the controls.

In this chapter, we study a leader-follower stochastic differential game with asymmetric information. Our work distinguishes itself from these mentioned above in the following aspects. (1) In our framework, the information available to the follower is based on some sub-*forward-backward stochastic differential equation* (FBSDE) with complete information for the leader. This problem is new in differential game theory and have considerable impacts in both theoretical analysis and practical meaning with future application prospect, although it has intrinsic mathematical difficulties. (4) The Stackelberg equilibrium of this LQ problem is characterized in terms of the *forward-backward stochastic differential filtering equations* (FBSDFEs) which arises naturally in our setup. These FBSDFEs are new and different from those in [10, 16]. (5) The Stackelberg equilibrium of this LQ problem is explicitly given, with the help of some new Riccati equations.

The rest of this chapter is organized as follows. In Section 2, we solve our problem to find the Stackelberg equilibrium. In Section 3, we apply our theoretical results to an LQ problem. Finally, Section 4 gives some concluding remarks.

## 2. Stackelberg equilibrium

### 2.1. The Follower’s problem

In this subsection, we first solve **SOCPF**. For any chosen

Let an

Proposition 2.1 *Let* **(A1.1)** *hold. For any given* *, let* *be the optimal control for* **SOCPF***, and* *be the corresponding optimal state. Let* *be the adjoint process triple. Then we have*

holds, for any

*Proof* Similar to the proof of Theorem 2.1 of [10], we can get the result.

Proposition 2.2 *Let* **(A1.1)** *hold. For any given* *, let* *and* *be the corresponding state. Let* *be the adjoint process triple. For each* *,* *is concave,* *is convex, and*

holds for **SOCPF**.

*Proof* Similar to the proof of Theorem 2.3 of [10], we can obtain the result.

### 2.2. The Leader’s problem

In this subsection, we first state the **SOCPL**. Then, we give the maximum principle and verification theorem. For any

For the simplicity of notations, we denote

Note that Eq. (26) is a controlled *conditional mean-field FBSDE*, which now is regarded as the “state” equation of the leader. That is to say, the state for the leader is the quadruple

Remark 2.1 The equality

Define

where

**SOCPL.** Find a

subject to Eqs. (26) and (27). Such a

Let

Let

Now, we have the following two results.

Proposition 2.3 *Let* **(A1.1)** *hold. Let* *be an optimal control for* **SOCPL** *and* *be the optimal state. Let* *be the adjoint quadruple, then*

*Proof* The maximum condition Eq. (31) can be derived by convex variation and adjoint technique, as Anderson and Djehiche [17]. We omit the details for saving space. See also Li [18], Yong [19] and the references therein for mean-field stochastic optimal control problems.

Proposition 2.4 *Let* **(A1.1)** *hold. Let* *and* *be the corresponding state, with* *. Let* *be the adjoint quadruple. For each* *, suppose that* *and* *are convex, and*

Then **SOCPL**.

*Proof* This follows similar to Shi [20]. We omit the details for simplicity.

## 3. Applications to LQ case

In order to illustrate the theoretical results in Section 2, we study an LQ leader-follower stochastic differential game with asymmetric information. In this section, we let

### 3.1. Problem of the follower

Suppose that the state

Here,

In the second step, knowing that the follower would take

where

Define the Hamiltonian function of the follower as

For given control

where the

We wish to obtain the state feedback form of

for some deterministic and differentiable

In the above equation,

Comparing Eq. (41) with Eq. (38), we arrive at

and

respectively. Taking

and

respectively. Applying Lemma 5.4 in [21] to Eqs. (33) and (38) corresponding to

Note that Eq. (46) is not a classical FBSDFE, since the generator of the BSDE depends on an additional process

**(A2.1)** *,* *,*

we immediately arrive at

where

admits a unique differentiable solution

where **(A2.1)**, we know that Eq. (48) admits a unique solution

which recovers the standard one in [22]. With Eq. (49), the BSDE Eq. (40) takes the form

Moreover, for given

we have

which admits a unique

where

which admits a unique

Theorem 3.1 *Let* **(A2.1)** *hold,* *satisfy* Eq. (48)*. For chosen* *of the leader,* *given by* Eq. (47) *is the optimal control of the follower, where* *is the unique* *-adapted solution to* Eq. (55).

### 3.2. Problem of the leader

Since the leader knows that the follower will take

where

The problem of the leader is to choose an

is minimized. Define the Hamiltonian function of the leader as

Suppose that there exists an

where the

In fact, the problem of the leader can also be solved by a direct calculation of the derivative of the cost functional. Without loss of generality, let

Hence

Let the

Applying Itô’s formula to

This implies Eq. (59).

In the following, we wish to obtain a “nonanticipating” representation for the optimal controls

and (suppressing some

With the notations, Eq. (56) with Eq. (60) is rewritten as

Noting Eq. (59), we have

Inserting Eq. (68) into Eq. (67), we get

where

We need to decouple Eq. (69). Similar to Eq. (39), put

where

Applying Itô’s formula to (3.31), we get

Comparing

Taking

Supposing that (

we get

where

Inserting Eq. (77) into Eq. (74), we have

Supposing that

we get

where.

Comparing the coefficients of

Note that the system of Riccati equations (83) is not standard, and its solvability is open. Due to some technical reason, we can not obtain the solvability of it now. However, in some special case,

Instituting Eqs. (77) and (81) into Eq. (68), we obtain

and the optimal “state”

where

We summarize the above analysis in the following theorem.

Theorem 3.2 *Let* **(A2.1)****(A2.3)** *hold,* *satisfy* Eq. (83)*,* *be the* *-adapted solution to* Eq. (86)*, and* *be the* *-adapted solution to* Eq. (85)*. Define* *by* Eqs. (71) *and* (81)*, respectively. Then* Eq. (69) *holds, and* *given by* Eq. (84) *is a feedback optimal control of the leader.*

Finally, the optimal control

which is observable for the follower.

**Remark 3.3** When we consider the complete information case, that is,

## 4. Concluding remarks

In this chapter, we have studied a leader-follower stochastic differential game with asymmetric information. This kind of game problem possesses several attractive features. First, the game problem has the Stackelberg feature, which means the two players play as different roles during the game. Thus the usual approach to deal with game problems, such as [6, 7, 8, 10], where the two players act as equivalent roles, does not apply. Second, the game problem has the asymmetric information between the two players, which was not considered in [3, 13, 14]. In detail, the information available to the follower is based on some sub-

In principle, Theorems 3.1 and 3.2 provide a useful tool to seek Stackelberg equilibrium. As a first step in this direction, we apply our results to the LQ problem to obtain explicit solutions. We hope to return to the more general case in our future research. It is worthy to study the closed-loop Stackelberg equilibrium for our problem, as well as the solvability of the system of Riccati equations. These challenging topics will be considered in our future work.

## Acknowledgments

Jingtao Shi would like to thank the book editor for his/her comments and suggestions. Jingtao Shi also would like to thank Professor Guangchen Wang from Shandong University and Professor Jie Xiong from Southern University of Science and Technology, for their effort and discussion during the writing of this chapter.

## Notes

The main content of this chapter is from the following two published article papers: (1) Shi, J.T., Wang, G.C., & Xiong, J. (2016). Leader-follower stochastic differential games with asymmetric information and applications. *Automatica*, Vol. 63, 60–73. (2) Shi, J.T., Wang, G.C., & Xiong, J. (2017). Linear-quadratic stochastic Stackelberg differential game with asymmetric information. *Science China Information Sciences*, Vol. 60, 092202:1–15.