An Open-Loop Stackelberg Strategy for the Linear Quadratic Mean-Field Stochastic Differential Game

Lin, Yaning; Jiang, Xiushan; Zhang, Weihai

doi:10.1109/tac.2018.2814959

Cited by 90 publications

(61 citation statements)

References 31 publications

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“…Note that the problem formulation and the results of the paper can be viewed as extensions of those in [40], [50] to the time-inconsistent problem, and those in [3], [25] to the Stackelberg game framework. In [50], the LQ mean-field Stackelberg game was considered, where the corresponding Stackelberg equilibrium is precommitted, i.e., it is time inconsistent. The extensions of [40], [50] to the time-inconsistent setting are not trivial, since the approach for characterizing the (time-consistent) equilibrium control is completely different from that of the the precommitted (time-inconsistent) optimal solution as discussed in [1], [3], [26].…”

Section: A Problem Statement and Main Results Of The Papermentioning

confidence: 99%

“…In [50], the LQ mean-field Stackelberg game was considered, where the corresponding Stackelberg equilibrium is precommitted, i.e., it is time inconsistent. The extensions of [40], [50] to the time-inconsistent setting are not trivial, since the approach for characterizing the (time-consistent) equilibrium control is completely different from that of the the precommitted (time-inconsistent) optimal solution as discussed in [1], [3], [26]. On the other hand, in [3] the time-inconsistent mean-field control problem was studied, where the (adapted open-loop time-consistent) equilibrium control was obtained.…”

Section: A Problem Statement and Main Results Of The Papermentioning

confidence: 99%

“…In (2) and (3), the cost parameters depend on the initial time t. In addition, not only the state and control variables, but also their conditional expectations are included nonlinearly in the objective functionals in (2) and (3). The conditional expectation terms in (2) and (3) are also known as mean field of state and control variables [3], [25], [30], [31], [50]. The interaction between the leader and the follower can be stated as follows.…”

Section: Problem Statementmentioning

confidence: 99%

“…3 In particular, for the adapted open-loop case, the (stochastic) Stackelberg game can be analyzed by applying the stochastic maximum principle to the following: (i) the follower's problem given an arbitrary strategy of the leader and (ii) the leader's problem, where the constraint is the follower's rational behavior characterized by the forward-backward SDE (FBSDE) from (i) [39]- [41]. There are wide ranges of applications for Stackelberg games including engineering, economics and biology; see [29], [46]- [50] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

Moon

Yang

2021

IEEE Trans. Automat. Contr.

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In this paper, we consider the linear-quadratic timeinconsistent mean-field leader-follower Stackelberg stochastic differential game with an adapted open-loop information structure. Given a controlled linear stochastic differential equation (SDE), the quadratic objective functionals of the leader and the follower include conditional expectations of state and control (mean field) variables. In addition, the cost parameters could be general nonexponential discounting depending on the initial time. As stated in the existing literature, these two general settings of the objective functionals induce time inconsistency in the optimal solutions. Given an arbitrary control of the leader, we first obtain the follower's (time-consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled Riccati differential equations (RDEs) and the backward SDE. This provides the rational behavior of the follower, characterized by the forward-backward SDE (FBSDE). We then obtain the leader's explicit (time-consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled RDEs, where the constraint of the leader's problem is the FBSDE induced by the follower's rational behavior. With the solvability of the nonsymmetric coupled RDEs, the equilibrium controls of the leader and the follower constitute the time-consistent Stackelberg equilibrium of the paper. Finally, the numerical examples are provided to check the solvability of the nonsymmetric coupled RDEs of the leader and the follower.Index Terms-time-inconsistent stochastic control problem, equilibrium control, Stackelberg differential games, mean-field stochastic systems. 4 This implies that the magnitude of the stochastic noise is controlled by the leader and the follower, which can be viewed as multiplicative noise of the system [51], [52]. 5 As mentioned, an equilibrium control is time consistent; see Definition 2 or [3, Definition 4.1] and [26, Definition 2.1].

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Section: A Problem Statement and Main Results Of The Papermentioning

confidence: 99%

Section: A Problem Statement and Main Results Of The Papermentioning

confidence: 99%

Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

Moon

Yang

2021

IEEE Trans. Automat. Contr.

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“…Besides, Wu and Liu [19] studied an optimal control problem for mean-field zero-sum stochastic differential game under partial information. Recently, Lin et al [20] discussed an open-loop LQ leader-follower of meanfield stochastic differential game and solved the corresponding optimal control problems for the follower and the leader; Du and Wu [21] considered a new kind of Stackelberg differential game of MF-BSDE, and they obtained the open-loop Stackelberg equilibrium, which admits a state feedback representation. What is more, Zhang [22] investigated an optimal control problem for terminal constraint mean-field SDE under partial information, which was solved by the backward separation method with a decomposition technique.…”

Section: Introductionmentioning

confidence: 99%

An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications

2019

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This paper is concerned with a kind of non-zero sum differential game driven by mean-field backward stochastic differential equation (MF-BSDE) with asymmetric information, whose novel feature is that both the state equation and the cost functional are of mean-field type. A necessary condition and a sufficient condition for Nash equilibrium point of the above problem are established. As applications, a mean-field linear-quadratic (MF-LQ) problem and a financial problem are studied.

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Feasible solution to discrete‐time linear quadratic stochastic Stackelberg difference game

Qi,

Zhang,

Sun

2023

Asian Journal of Control

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This paper investigates the discrete‐time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward‐backward stochastic differential equations (SG‐FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.

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An Open-Loop Stackelberg Strategy for the Linear Quadratic Mean-Field Stochastic Differential Game

Cited by 90 publications

References 31 publications

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications

Feasible solution to discrete‐time linear quadratic stochastic Stackelberg difference game

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