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.
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
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
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
subject to Eqs. (13) and (15). Such a
In the following step, once knowing that the follower will take such an optimal control
Here
subject to Eqs. (13) and (17). Such a
In this chapter, we impose the following assumptions.
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-
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
Let an
Proposition 2.1
holds, for any
Proposition 2.2
holds for
2.2. The Leader’s problem
In this subsection, we first state the
For the simplicity of notations, we denote
Note that Eq. (26) is a controlled
Remark 2.1 The equality
Define
where
subject to Eqs. (26) and (27). Such a
Let
Let
Now, we have the following two results.
Proposition 2.3
Proposition 2.4
Then
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
we immediately arrive at
where
admits a unique differentiable solution
where
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
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
Finally, the optimal control
which is observable for the follower.
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.
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