Stackelberg Strategy Research Articles

Optimal Control and Filtering for Hierarchical Decision Problems With Constraint Based on Stackelberg Strategy

Optimal Control and Filtering for Hierarchical Decision Problems With Constraint Based on Stackelberg Strategy

Fairness mechanism and stackelberg strategy for multi-agent systems: A case study of legends of the three kingdoms

Alongside the rapid advancement of artificial intelligence technologies, the complexity of society continues to escalate, and the design of multi-agent system rules becomes increasingly crucial, optimize participant engagement and addressing the issue of fairness and equilibrium win rates. The card game Legends of the Three Kingdoms (LTK), as a complex multi-agent system, represents an important abstraction of real-world scenarios, requiring the implementation of proper incentive mechanisms. Based on the mechanism and Stackelberg, this paper optimizes player strategies and system fairness. Firstly, we constructed a multi-agent Stackelberg game model for team battles. Then, we analyzed the impact of decision-making factors on players. Subsequently, we defined three kinds of fairness and improved the incentive mechanism. The results indicate that our mechanism optimizes participants' game behaviors, enhancing fairness among team players. Win rates were improved from 56.2% VS 43.8% to 51.4% VS 48.6%. With three different fairness measures, the win percentage fairness increased by 72 %, and the first elimination fairness increased by about 79 %. Our research provides a reference for understanding and analyzing complex computational models and facilitates the resolution of various resource allocation and system design issues.

Hierarchical control problem for the heat equation with dynamic boundary conditions

Abstract This paper deals with a hierarchical control problem for the heat equation with dynamic boundary conditions. The main goal consists of letting the state near from a prescribed target in a fixed observation region. The secondary objective is the null controllability. In other words, we reverse the roles of the leader and the follower addressed in the recent article. For this purpose, we combine some appropriate Carleman estimates and the Stackelberg strategy. We also extend our study for hierarchical-biobjective problems by applying the Stackelberg–Pareto strategy.

Impact of uncertain demand and channel power on dual channel supply chain decisions.

The paper aims to conduct an analysis of pricing strategies in a dual channel supply chain under external uncertainty, utilizing Interval numbers theory and Game theory as the theoretical basis. The focus is on maximizing the expected profits of manufacturers and retailers. Four models are considered: centralized decision-making, manufacturer's Stackelberg, retailer's Stackelberg strategy, and vertical Nash model, with the decision variable being the product price. By solving the game model, the paper compares the optimal decisions under the four models and conducts sensitivity analysis to reflect the influence of key parameters and analyze their relationships. The ultimate goal is to optimize profits under various circumstances by adjusting market potential and price parameters to determine the best price level. The findings suggest that decision-maker's risk indicators have a greater impact on decision results when market demand is less sensitive to price, and that the size of the market has a negative correlation with the impact of decision-maker's risk indicators on decision results.

Adaptive robust control for fuzzy underactuated mechanical systems: A Stackelberg game-theoretic optimization approach

This article proposes a Stackelberg game-theoretic optimization approach for adaptive robust controller design of fuzzy underactuated mechanical systems (UMSs). As a commonly encountered challenge, several factors render it difficult to further improve the control performance of these UMSs, such as the coupling effects and nonlinearities resulting from the underactuated structure. Meanwhile, various sources of unexpected time-varying uncertainties can further add to the difficulty in the controller design procedure pipeline. To overcome the above-mentioned issues and further enhance the system performance, a dual-parameter optimization framework is presented here for adaptive robust controller design, leveraging fuzzy set theory and Stackelberg game. In this development, the time-varying uncertainty is characterized via suitable membership functions; and thus the appropriate fuzzy UMS is established. Then, an adaptive robust controller with a leakage-type adaptation law is presented to tackle the uncertainty appropriately. To further improve the system performance considering the trade-off among the transient performance; the steady-state performance; and the control cost; a Stackelberg game-based dual-parameter optimization method is proposed to determine the Stackelberg strategy. It is rigorously proved in the developments here that the proposed controller design and parameters optimization method guarantees the desired deterministic system performance, i.e., uniform boundedness (UB) and uniform ultimate boundedness (UUB). Moreover, a series of numerical simulations based on a flywheel inverted pendulum (FIP) are performed, which illustrate the effectiveness and superiority of the proposed method.

Stackelberg exact controllability of a class of nonlocal parabolic equations

This paper deals with a multi-objective control problem for a class of nonlocal parabolic equations, where the non-locality is expressed through an integral kernel. We present the Stackelberg strategy that combines the concepts of controllability to trajectories with optimal control. The strategy involves two controls: a main control (the leader) and a secondary control (the follower). The leader solves a controllability to trajectories problem which consists to drive the state of the system to a prescribed target at a final time while the follower solves an optimal control problem which consists to minimize a given cost functional. The paper considers two cases: in the first case, both the leader and the follower act in the interior of the domain, and in the second case, the leader acts in the interior of the domain and the follower acts on a small part of the boundary. These results are applied to both linear and nonlinear nonlocal parabolic systems.

Active Interdiction Defence Scheme Against False Data-Injection Attacks: A Stackelberg Game Perspective.

The security issue is of the fundamental importance for cyber-physical systems (CPSs) due to the vulnerability to cyber attacks. In this article, an active interdiction defence scheme is proposed as an extra safeguard against the strategic false data-injection (FDI) attacker. To tackle the coupling between the decision-making processes of the FDI attacker and the active interdiction defence scheme, a framework of dynamic Stackelberg game is formulated to characterize the hierarchical interaction due to their asymmetric information structures. Subsequently, the Stackelberg strategies of both players are constructed. Furthermore, a sufficient condition is derived to guarantee asymptotic stability of the closed-loop system with the FDI attack and the active interdiction defence scheme. Finally, two simulation examples are provided to illustrate the correctness and effectiveness of the proposed active interdiction defence scheme.

Robust control design for Electric Helicopter Tail Deceleration system: Fuzzy view and Stackelberg game theory-based optimization

A novel robust controller is designed for Electric Helicopter Tail Deceleration (EHTD) system which is subject to uncertainties and nonlinearity. Then, the optimal parameters of the control are solved based on Stackelberg game theory. The fuzzy set theory is used to describe the uncertainty bound, which is distinguished from probability theory or fuzzy logic theory. Based on two-player (leader and follower) Stackelberg game, two performance indexes are proposed for novel robust controller which contains two optimal parameters. The system performance and control cost are included in performance indexes. Then, based on Stackelberg strategy, the optimal decisions are made by leader and follower, which can minimize the performance indexes. Therefore, by numerical simulation, the EHTD system is applied to prove that the design of robust control and optimal parameters based Stackelberg strategy are effective.

Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information

Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information

A Stackelberg Game Approach to the Stability of Networked Switched Systems Under DoS Attacks

This paper investigates the stability of networked switched systems under DoS attacks by a Stackelberg game approach. Firstly, the non-cooperative competition between the attacker and the controller is characterized as a switched Stackelberg game. The delivery rate of data packets and controller gains compose the strategy sets of the attacker and the controller, respectively. Then, the switched optimal Stackelberg strategy is obtained from the corresponding Stackelberg equilibrium. Considering the open-loop and asynchronous switching cases caused by DoS attacks, sufficient conditions for the exponential mean-square stability of the system are obtained. Finally, switched RLC circuit systems are provided to verify the effectiveness of the proposed methods.

A Stackelberg Game-Theoretic Exploration Rendering Robustness and Optimality for Performance Improvement of Fuzzy Mechanical Systems.

We consider mechanical systems with uncertainty. The uncertainty may be time varying. The bound of the uncertainty is described by its fuzzy characteristics. To design a feasible control, we start with a robust phase, which renders a control scheme that guarantees the system performance regardless of the actual value of the uncertainty. This robust phase is then followed by an optimal phase. There are design parameters in the control, which can be fine-tuned. We proposed multiple performance objectives. The goal of the choice of the control design parameters is to minimize the performance objectives. However, since these objectives are nonconciliating (meaning one's minimum is not the other one's minimum), we invoke the Stackelberg strategy for the optimal parameters. The game strategy mimics two players: one is the leader and one is the follower. Through the interplay between the two players, we show how to select the design parameters. The design procedure in both robust and optimal phases is demonstrated by a coupled inverted pendulum system.

A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game

<p style='text-indent:20px;'>In this paper, we consider a general form of linear-quadratic Stackelberg deterministic differential game model, which consists of one leader and one follower. Each of their utility functions includes all possible squared terms, cross terms and single terms of states and controls of the two players, and constant terms. The time-consistent state feedback form of Stackelberg equilibrium strategy is obtained. Its explicit expression is in terms of the solutions of three decoupled symmetric Riccati differential equations. These decoupled symmetric Riccati differential equations are independent of the state and can be solved backward in time one by one. The proposed model and theory are applied to some classical Stackelberg games.</p>

Coordination in a closed-loop sustainable supply chain considering dual-channel and cost-sharing contract: Evidence from an emerging economy

Due to the significant role of the reverse supply chain (RSCs) in collecting used products and achieving a sustainable environment, both scholars and industries have paid close attention to pricing in reverse and closed-loop supply chains (CLSCs). Moreover, with the rapid development of the Internet and e-commerce in the latest decades, researchers have examined the impact of constructing online return channels based on customer behavior. In this article, a game-theoretic approach was applied to find the optimal economic and environmentally sustainable solutions in a two-level CLSC with a dual collecting channel including the retailer’s traditional channel and the manufacturer’s online channel. The purpose of the current study is to optimise the selling price, acquisition prices, market demand, channels return rate, the portion of manufacturing new products, and cost-sharing contract (CSC) participation shares for each player. For this purpose, various policies, such as centralised and decentralised modes, different structures such as Nash bargaining power, manufacturer-leader Stackelberg, and retailer-leader Stackelberg have been considered. However, the main contribution of this work compared to the existing literature is considering two CSCs from both retailer and manufacturer points of view, with a real case analysis from an emerging economy. In addition, a comprehensive sensitivity analysis has been carried out to enhance the validation of the proposed model. The results indicated that the manufacturer-leader Stackelberg strategy leads to the lowest profit for the SC in both decentralised and cooperative policies. However, when the retailer and manufacturer have equal decision-making power (Nash strategy) and the retailer participates in the remanufacturing cost (i.e. cost-sharing type-2) both the economic and environmentally sustainable goals of CLSC were met.

Incentive feedback Stackelberg strategy for the discrete-time stochastic systems

The paper is devoted to designing an incentive Stackelberg strategy for the discrete-time stochastic systems. Most of the existing works are about the deterministic systems and the continuous-time stochastic systems. Although many methods have been used to help the leader to attain his team-optimal value, due to the involvement of the conditional expectation, the design of the incentive Stackelberg strategy for the discrete-time stochastic systems is more difficult and complicated. In this paper, this problem is studied for both the finite horizon case and the infinite horizon situation. Also, the mean-square stability is guaranteed by the follower. In addition, for the infinite horizon case, the specific algorithm procedure is put forward to obtain the incentive feedback Stackelberg strategy conveniently. At last, two examples are given to verify the effectiveness of the proposed algorithm procedure.

On the Computation of Hierarchical Control results for One-Dimensional Transmission Line

In this paper, motivated by a physics problem, we investigate some numerical and computational aspects for the problem of hierarchical controllability in a one-dimensional wave equation in domains with a moving boundary. Some controls act in part of the boundary and define a strategy of equilibrium between them, considering a leader control and a follower. Thus, we introduced the concept of hierarchical control to solve the problem and mapped the Stackelberg Strategy between these controls. A total discretization of the problem is presented for a numerical evaluation in spaces of finite dimension, an algorithm for evaluation of the problem is presented as the combination of finite element method (FEM) and finite difference method (FDM). The algorithm efficiency and computational results are illustrated for some experiments using the softwares Freefem++ and MatLab.

Stackelberg-Game-Oriented Optimal Control for Bounded Constrained Mechanical Systems: A Fuzzy Evidence-Theoretic Approach

This article proposes a novel Stackelberg-game-oriented optimal control approach to address the bounded constraint-following control problem for uncertain mechanical systems. First, the uncertainties (possibly fast time-varying) in the system are assumed to be bounded with an unknown boundary, which lies in a specified fuzzy evidence number. In practical engineering, bounded system performance is always demanded, such as the inequality constraint. A diffeomorphism transformation approach is proposed to transform the constrained system into a restructured one satisfying the bounded constraint. Second, we propose an adaptive robust control oriented by the constraint-following control to render the restructured system to follow the specified constraints accurately with deterministic performance (guaranteeing uniform boundedness and uniform ultimate boundedness). The self-adjusting adaptive law (leakage-type) can compensate for the uncertainties and avoid overcompensation. Third, a Stackelberg-game-oriented optimization approach is proposed to obtain the optimal control parameters based on the fuzzy evidence theory. In the optimization approach, the two control parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> are considered as two players with respective cost functions related to system performance and control cost. Furthermore, the optimization problem is solved by obtaining the Stackelberg strategy, which is proved to exist in analytic form. Ultimately, the permanent magnet synchronous linear motor system simulation is presented to show the design process and the excellent performance of the proposed optimal control scheme.

Incentive Stackelberg game for H∞$H_\infty$‐constrained multi‐hierarchy systems under observation information

This paper considers the incentive feedback Stackelberg game with multi-hierarchy players under a H ∞ $H_\infty$ constraint, with the proposed solution involving nested hierarchies A and B. In hierarchy A, P0 represents the leader (the leader's control input corresponding to i-th follower, i = 1 , 2 , … , n $i=1,2,\ldots ,n$ ), and P 1 , … , P n $P_1,\ldots ,P_n$ are the followers, with non-cooperative followers induced to virtually cooperate in achieving team-optimal solution and the Nash equilibrium. In hierarchy B, the external disturbance represents the follower, and hierarchy A represents the leader. The main contributions of this work are three-fold. First, an incentive Stackelberg strategy set under the H ∞ $H_\infty$ constraint through observation information is obtained for the first time. Second, a novel iterative algorithm to solve the coupled backward and forward Riccati equations is introduced and thus obtaining an explicit expression of a team-optimal feedback Stackelberg strategy set with an H ∞ $H_\infty$ constraint. Finally, the necessary and sufficient conditions for the existence and uniqueness of the hierarchical game's optimal solutions using the Lyapunov equation and the induction algorithm are presented.

Coordination of Collective Actions by Using the Stackelberg Strategy

Coordination of Collective Actions by Using the Stackelberg Strategy

Robust Stackelberg Differential Game With Model Uncertainty

This article formalizes two types of modeling uncertainties in a stochastic Stackelberg linear-quadratic (LQ) differential game and then discusses the associated robust Stackelberg strategy design for either the leader or follower. Both uncertainties are primarily motivated by practical applications in engineering and management. The first uncertainty is connected to a disturbance unknown to the follower but known to the leader. A <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">soft-constraint</i> min-max control is applied by the follower to determine the optimal response, and then an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">augmented</i> LQ forward-backward stochastic differential equation control is solved by the leader to ensure a robust strategy design. The second uncertainty involves a disturbance, the realization of which can be completely observed by the follower, but only its distribution can be accessed by the leader. Thus, a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hard-constraint</i> min-max control on an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">affine-equality-constraint</i> is studied by the leader to address the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">exact-optimal</i> robust design. Moreover, based on a weak convergence technique, a minimizing sequence of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">near-optimal</i> robust designs is constructed, which is more tractable in computation. Some numerical results of the abovementioned robust strategies are also presented.

Output‐feedback Q‐learning for discrete‐time linear H∞ tracking control: A Stackelberg game approach

AbstractIn this article, an output‐feedback Q‐learning algorithm is proposed for the discrete‐time linear system to deal with the tracking control problem. The problem is formulated as a zero‐sum game in the Stackelberg game framework with a discount factor to make the value function bounded. According to the principle of optimality, the game algebraic Riccati equation (GARE) is derived and solved by the Q‐learning algorithm to get the optimal solution of the Stackelberg game without requiring the knowledge of system dynamics and state. It is proved that the solution of the algorithm converges to the optimal control input and the worst‐case disturbance with excitation noises during training, and the Stackelberg strategy can achieve a lower disturbance attenuation level than the Nash one. Moreover, the impacts of the discount factor on the stability of the closed‐loop system and solvability of the GARE are analyzed to provide some criteria for the choice of the discount factor. Simulation examples are provided to validate the effectiveness of the algorithm.