Stackelberg Strategy Research Articles

Impact of Hostile Interference on Wireless Link Connectivity

We formulate the interaction between link connectivity and hostile interference in a wireless system as a nonzero-sum two-player game. One player is the transmitter aiming to maintain the connections to its receivers, whereas the other player is the interferer aiming to disrupt the connections. The players’ strategies are their transmitted power levels. We focus on the case in which the ratio of the averages of the channel power gains for the transmitter and interferer are the same at all receivers. We then provide closed-form expressions for the Nash and Stackelberg equilibria. When the channel is affected by Rayleigh fading, we show that both players have the same power cost at Nash equilibria. In addition, the Stackelberg strategies dominate the Nash strategies, that is, the utility functions of the Stackelberg equilibria exceed or equal those of the Nash equilibria.

Robust Stackelberg controllability for the Navier–Stokes equations

In this paper we deal with a robust Stackelberg strategy for the Navier–Stokes system. The scheme is based in considering a robust control problem for the “follower control” and its associated disturbance function. Afterwards, we consider the notion of Stackelberg optimization (which is associated to the “leader control”) in order to deduce a local null controllability result for the Navier–Stokes system.

A Stackelberg strategy for continuous-time mixed H2/H∞ control problem with time delay

This paper is concerned with the mixed H2/H∞ control with linear continuous time system and time delay. To deal with this, we presents a Stackelberg strategy by treating the control input and the disturbance as leader and follower, respectively. The leader’s control strategy minimizes the cost function which is in H2 norm and the follower’s control strategy maximizes the cost function which is in H∞ norm. The main technique of this paper is deal with the noncausal relationship of the variables caused by time delay in the control input by introducing two costates to capture the future information and one state to capture the past information. Through theory analyzing, the Stackelberg strategy exists uniquely. Moreover, with the assistance of the extended state space expression, the explicit expression of the strategy is obtained.

Incentive Stackelberg Games for Stochastic Linear Systems With H∞ Constraint.

In this paper, incentive Stackelberg games with one leader and multiple followers are investigated for a class of stochastic linear systems with external disturbance. Unlike the existing ordinary Stackelberg games, the leader is required to design an incentive Stackelberg strategy set that can lead to the leader's team-optimal solution and the follower's Nash equilibrium, and attenuate the external disturbance in the system simultaneously. It is shown that the incentive Stackelberg strategy set is obtained by solving a set of the cross-coupled stochastic Riccati differential equations in a finite-horizon case and a set of the cross-coupled stochastic algebraic Riccati equations in an infinite-horizon case. Numerical examples are solved to demonstrate the effectiveness of the proposed incentive Stackelberg strategy set.

Robust Stackelberg controllability for linear and semilinear heat equations

In this paper, we present a Stackelberg strategy to control a semilinear parabolic equation. We use the concept of hierarchic control to combine the concepts of controllability with robustness. We have a control named the leader which is responsible for a controllability to trajectories objective. Additionally, we have a control named the follower, that solves a robust control problem. That means we solve for the optimal control in the presence of the worst disturbance case. In this way, the follower control is insensitive to a broad class of external disturbances.

Virtual Stackelberg game coupled with the adjoint method for aerodynamic shape optimization

ABSTRACTIn this article, a Virtual Stackelberg Game (VSG) is proposed for aerodynamic shape optimization, where the design variables are divided into two categories to optimize the same objective function, one acts as a leader, and the other ones as followers react independently and selfishly relative to the leader's strategy. During each Stackelberg strategy cycle, the Gradient-Based Method (GBM) with the adjoint method in Stanford University Unstructured (SU2) is applied in the optimization of each player. Firstly, parametric studies of VSG by two simple cases are conducted to assess the impact of critical parameters on aerodynamic shape optimization, including the design cycle, the split of design variables and role (leader and follower) assignment. Based on the criterion from parametric studies, two typical numerical cases under transonic flow are applied—the drag reduction design of a 2D airfoil and a 3D wing. It is found that, compared to the original GBM method, VSG can provide better optimization results with less computational cost.

Optimal Stackelberg strategies for financing a supply chain through online peer-to-peer lending

In recent years, supply chain finance (SCF) through online peer-to-peer (P2P) lending platforms has gained its popularity. We study an SCF system with a manufacturer selling a product to a retailer that faces uncertain demand over a single period. We assume that either the retailer or the manufacturer faces a capital constraint and must borrow capital through an online P2P lending platform. The platform determines a service rate for the loan, the manufacturer sets a wholesale price for the product, and the retailer chooses its order quantity for the product. We identify optimal Stackelberg strategies of the participants in the SCF system. For an SCF system with a capital-constrained retailer, we find that the retailer's optimal order quantity and the manufacturer's optimal wholesale price decrease with the platform's service rate. For an SCF system with a capital-constrained manufacturer, we find that as the platform's service rate increases, the manufacturer's optimal wholesale price increases but the retailer's optimal order quantity decreases. Our analysis suggests that it is important for the retailer and the manufacturer to take the online P2P lending platform's financial decisions (such as the service rate) into account when making their operational decisions.

‐constrained incentive Stackelberg games for discrete‐time stochastic systems with multiple followers

The authors discuss an incentive Stackelberg game with one leader and multiple non-cooperative followers, for a class of discrete-time stochastic systems with an external disturbance. In this game, the leader achieves a team-optimal solution by attenuating the external disturbance under their H ∞ constraint, whereas the followers adopt Nash equilibrium strategies according to the leader's incentive Stackelberg strategy set (declared in advance) while considering the worst-case disturbance. Using our proposed method, we demonstrate that the incentive Stackelberg strategy set can be found by solving a set of matrix-valued equations. Techniques are presented for both the finite- and infinite-horizon cases. In addition, through an academic and a practical numerical examples, we verify the efficacy of the proposed method in providing the incentive Stackelberg strategy set.

A bi-level programming approach to the decision problems in a vendor-buyer eco-friendly supply chain

This paper focuses on a multi-product vendor-buyer supply chain considering environmental factors in the product manufacturing process. We assume that the unit price charged by the buyer, the unit advertising expenditure and the vendor’s unit environmental improvement influence the demand of the product being sold. The relationship between the vendor and the buyer is modeled by a non-cooperative game based on Stackelberg strategy framework under two power scenarios including vendor-leader model and buyer-leader model. A bi-level programming approach is applied to determine the optimal selling prices, advertising expenditures, wholesale prices, vendor’s environmental improvements and ordering policies of the vendor and the buyer. Then, a modified differential evolution algorithm is proposed for solving the models. Numerical experiments carried out in this paper, including sensitivity analysis for some key parameters, evaluate the effectiveness of the models and compare the results between the different models considered. Several research findings have been obtained and explained.

Infinite horizon linear-quadratic Stackelberg games for discrete-time stochastic systems

In this paper, we consider infinite horizon linear-quadratic Stackelberg games for a discrete-time stochastic system with multiple decision makers. Necessary conditions for the existence of the Stackelberg strategy set are derived in terms of the solvability of cross-coupled stochastic algebraic equations (CSAEs). As an important application, the hierarchical H∞-constraint control problem for the infinite horizon discrete-time stochastic system with multiple channel inputs is solved using the Stackelberg game approach. Computational methods for solving the CSAEs are also discussed. A numerical example is provided to demonstrate the usefulness of the proposed algorithms.

Cournot–Stackelberg games in competitive delocation

In order to mitigate the effects of the contraction in demand during economic crises, firms face the need to reduce the number of facilities in their networks. This reduction must be conducted taking into consideration both the possible actions of their rival firms and the reaction of the affected customers, so that the loss of market share is minimised. In this article, we analyse the facility closing problem of two firms operating in a duopolistic market. The problem is modelled as a non-cooperative game over a binary integer programming formulation of the firms’ delocation problem. The possible outcome of the game is analysed for three different competitive scenarios: Myopic behaviour, Cournot conjectures, and Stackelberg strategies. These scenarios are analysed under the assumption that customers show certain level of loyalty to the firm that they initially resorted to. This assumption entitles us to establish the existence of Nash equilibria in the delocation game by means of the introduction of a social planner. Moreover, this social planner provides a mechanism for computing the Nash equilibrium in the Cournot delocation game. Additionally, we develop an algorithmic approach that provides a simple mechanism for finding equilibria under Stackelberg strategies. Experimental results indicate that for small reductions in the network size the solution under myopic behaviour is often an equilibrium. For large reductions, evidence has been found that there is a first mover advantage in the Stackelberg delocation game.

H∞-Constrained Incentive Stackelberg Game for Discrete-Time Systems with Multiple Non-cooperative Followers

Abstract: In this paper, an H ∞ -constrained team-optimal strategy of an incentive Stackelberg game for a class of discrete-time system with an external disturbance is investigated. The team-optimal solution for the leader is achieved in contrast to multiple non-cooperative followers’ optimal state feedback gain. The solution sets for incentive Stackelberg strategy are found by solving a set of backward difference Riccati equations (BDREs) in the finite-horizon case. On the other hand, it is shown that the results for the infinite-horizon case are found by solving a set of algebraic Riccati equations (AREs). In particular, it is indicated that the team optimal solutions are yielded even though noncooperative strategy is considered for the multiple followers. In order to demonstrate the effectiveness of the proposed method, a numerical example is given.

Sufficient and Necessary Open-Loop Stackelberg Strategy for Two-Player Game With Time Delay.

This paper is concerned with both difference and differential leader-follower games with time delay. The problem remains challenging although the leader-follower game for delay-free system has been well studied in the past decades. The main obstacle encountered is the noncausality of strategy design caused by the delay. The key technique developed to overcome the difficulty is the introduction of the new co-states which capture the future information of the control and the new state which contains the past effects. The novel contributions of this paper are as follows. First, the sufficient and necessary solvability condition is given to ensure the existence of a unique open-loop Stackelberg strategy for the difference game. Second, the open-loop strategy is explicitly obtained in terms of decoupled and symmetric Riccati equations. Last but not least, a unique strategy for continuous time systems is also given by applying the techniques developed in this paper.

Necessary and Sufficient Condition for Two-Player Stackelberg Strategy

This technical note revisits the open-loop Stackelberg strategy for a two-player game. By introducing a new costate, which captures the future information of the control input, we present a necessary and sufficient condition for the existence and uniqueness of the two-player game. The optimal strategy is designed in terms of three decoupled and symmetric Riccati equations which improves the existing results greatly on computation.

Remarks on Hierarchic Control for a Linearized Micropolar Fluids System in Moving Domains

We study a Stackelberg strategy subject to the evolutionary linearized micropolar fluids equations in domains with moving boundaries, considering a Nash multi-objective equilibrium (non necessarily cooperative) for the "follower players" (as is called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following main results: the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability of the linearized micropolar system with respect to the leader control and the existence and uniqueness of the Stackelberg---Nash problem, where the optimality system for the leader is given.

Stackelberg strategies for stochastic systems with multiple followers

In this paper, Stackelberg games for linear stochastic systems governed by Itô differential equations with multiple followers are investigated. Stackelberg strategies are developed under two different settings, that is, the followers act either cooperatively to attain Pareto optimality or non-cooperatively to arrive at a Nash equilibrium. After establishing the necessary conditions to obtain Stackelberg strategies by using sets of cross-coupled algebraic nonlinear matrix equations (CANMEs), two numerical algorithms are developed based on Newton’s method and the semidefinite programming (SDP) problem to solve the CANMEs. Finally, a numerical example is solved to demonstrate the validity of the algorithms.

STOCHASTIC MARKSMANSHIP CONTEST GAMES WITH RANDOM TERMINATION — SURVEY AND APPLICATIONS

The marksmanship contest game with random termination is a fundamental two-person game of timing under an uncertain environment. In this paper, we formulate a unied marksmanship contest game with random termination and show that it can be further classied intove different games. We derive the optimal solutions of these games of timing, which are categorized into the Nash equilibrium strategy and the Stackelberg strategy. Our results involve the well-known results by Teraoka (1983) and Baston and Garnaev (1995) as special cases and at the same time give new insights for different types of game.

Upper Bounds on Inefficiency of Taxed Stackelberg Network with Cross- Nested Logit Assignment

Upper bound of efficiency loss is a valuable issue for transport network design and planning. This paper ini- tially explores it in a taxed stochastic traffic network whose equilibrium flow pattern is deduced by a cross-nested Logit (CNL) flow assignment model, and a centrally controlling Stackelberg strategy. With the assumptions of separability, nondecreasingness, and convexity of the link time function and the fixed origin-destination (OD) demand of network, the equivalent variational inequality for a CNL-based stochastic user equilibrium (CNL-SUE) model is established and first used to obtain upper bounds on Stackelberg network inefficiency. Further, for low-degree link time function such as Bu- reau of Public Roads and the affine forms, their inefficiency upper bounds are analyzed with some meaningful results.

Stackelberg strategies for wastewater management

Management of wastewater has become a critical environmental problem during past decades, specially for the particular case of urban areas. In this work we use numerical simulation, and combine optimal control theory of partial differential equations with multi-objective optimization techniques to formulate and solve this type of problems from a new original hierarchical viewpoint with a leader and a follower (Stackelberg strategies). Specifically, the main aim of this paper deals with applying these Stackelberg techniques to the environmental problem related to determining the optimal location and the optimal purification profile when building a new treatment plant in a wastewater depuration system. We analyze the control problem, introduce a numerical algorithm, and show the possibilities of our approach by solving a realistic problem posed in the Estuary of Vigo (NW Spain).

A Decomposition Algorithm for Solving a Three Level Large Scale linear Programming Problem

This paper presents a three level large scale linear programming problem in which the objective functions at every level are to be maximized. A three level programming problem can be thought as a static version of the Stackelberg strategy. An algorithm for solving a three planner model and a solution method for treating this problem are suggested. At each level we attempt to optimize its problem separately as a large scale programming problem using Dantzig and Wolfe decomposition method. Therefore, we handle the optimization process through a series of sub problems that can be solved independently. Finally, a numerical example is given to clarify the main results developed in this paper.