Michael Herty, Sonja Steffensen, Anna Thünen
We present a linear--quadratic Stackelberg game with a large number of followers and we also derive the mean field limit of infinitely many followers. The relation between optimization and mean-field limit is studied and conditions for consistency are established. Finally, we propose a numerical method based on the derived models and present numerical results.
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The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity conditions establish the existence theorem.
This paper is concerned with a Stackelberg stochastic differential game on a finite horizon in feedback information pattern. A system of parabolic partial differential equations is obtained at the level of Hamiltonian to give the verification theorem of the feedback Stackelberg equilibrium. As an example, a linear quadratic Stackelberg stochastic differential game is investigated. Riccati equations are introduced to express the feedback Stackelberg equilibrium, analytical and numerical solutions to these Riccati equations are discussed in some special cases.
The paper is concerned with a zero-sum Stackelberg stochastic linear-quadratic (LQ, for short) differential game over finite horizons. Under a fairly weak condition, the Stackelberg equilibrium is explicitly obtained by first solving a forward stochastic LQ optimal control problem (SLQ problem, for short) and then a backward SLQ problem. Two Riccati equations are derived in constructing the Stackelberg equilibrium. An interesting finding is that the difference of these two Riccati equations coincides with the Riccati equation associated with the zero-sum Nash stochastic LQ differential game, w…
In this paper, we consider a discrete-time stochastic Stackelberg game with a single leader and multiple followers. Both the followers and the leader together have conditionally independent private types, conditioned on action and previous state, that evolve as controlled Markov processes. The objective is to compute the stochastic Stackelberg equilibrium of the game where the leader commits to a dynamic strategy. Each follower's strategy is the best response to the leader's strategies and other followers' strategies while the each leader's strategy is optimum given the followers play the best…
A Fast and High-Accuracy Finite-Time Sliding-Mode Algorithm for Computing Local Stackelberg Equilibria in Nonlinear Bilevel Games
本文提出SSMA算法，将斯塔克尔伯格均衡计算转化为流形稳定问题，在500次非凸蒙特卡洛试验中实现100%成功率，迭代效率远优于基线，可快速高精度求解非线性双层博弈的局部斯塔克尔伯格均衡。
We consider a controlled linear-quadratic (LQ) large-population system with mixture of three types agents: major leader, minor leaders and minor followers. The Stackelberg-Nash-Cournot (SNC) approximate equilibrium is studied by a major-minor mean-field game (MFG) coupled with a leader-follower Stackelberg game. By variational method, the SNC approximate equilibrium strategy can be represented by some forward-backward-stochastic-differential-equations (FBSDEs) in the open-loop sense. And we pay great effort to give the feedback form of the open-loop strategy by some Riccati equations.
This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Finally, two examples are given to illustrate the applications of the theoretical results.
This paper proposes and studies a class of discrete-time finite-time-horizon Stackelberg mean-field games, with one leader and an infinite number of identical and indistinguishable followers. In this game, the objective of the leader is to maximize her reward considering the worst-case cost over all possible $ε$-Nash equilibria among followers. A new analytical paradigm is established by showing the equivalence between this Stackelberg mean-field game and a minimax optimization problem. This optimization framework facilitates studying both analytically and numerically the set of Nash equilibri…
Stackelberg games with the third party
本文在离散时间、动态和平均场三类Stackelberg博弈中引入第三方以实现时间不一致的Stackelberg均衡，通过类似行业协会的第三方利用折扣因子对领导者背叛行为施加惩罚，推导出防止背叛的条件，证明了该干预方法的有效性与可操作性。
This paper obtains the maximum principle for both stochastic (global) open-loop and stochastic (global) closed-loop Stackelberg differential games. For the closed-loop case, we use the theory of controlled forward-backward stochastic differential equations to derive the maximum principle for the leader's optimal strategy. In the special case of the open-loop linear quadratic Stackelberg game, we consider the follower's Hamiltonian system as the leader's state equation, derive the related stochastic Riccati equation, and show the existence and uniqueness of the solution to the Riccati equation …
In this paper, the known deterministic linear-quadratic Stackelberg game is revisited, whose open-loop Stackelberg solution actually possesses the nature of time inconsistency. To handle this time inconsistency, {a two-tier game framework is introduced, where the upper-tier game works according to Stackelberg's scenario with a leader and a follower, and two lower-tier intertemporal games give the follower's and leader's equilibrium response mappings that mimic the notion of time-consistent open-loop equilibrium control in existing literature. The resulting open-loop equilibrium solution of the…
Regulation of Rumor Propagation via (Multi-Leader) Stackelberg Graphon Games
本文研究通过斯塔克尔伯格图论游戏控制大规模网络中谣言传播，提出多领导者模型并设计双层算法，展示竞争性主体会导致群体意见分裂。
This paper is concerned with a Stackelberg stochastic differential game with asymmetric noisy observation, with one follower and one leader. In our model, the follower cannot observe the state process directly, but could observe a noisy observation process, while the leader can completely observe the state process. Open-loop Stackelberg equilibrium is considered. The follower first solve an stochastic optimal control problem with partial observation, the maximum principle and verification theorem are obtained. Then the leader turns to solve an optimal control problem for a conditional mean-fie…
Branch-and-price with novel cuts, and a new Stackelberg Security Game
该研究针对关键基础设施防护的Stackelberg安全博弈问题，提出新型有效不等式求解强Stackelberg均衡，结合分支定价法验证了方法优于现有标准公式，并分析了参数对性能的影响。
Closed-Loop Stackelberg Strategy for Linear-Quadratic Leader-Follower Game
本文针对线性二次领导者-跟随者博弈，利用约束最大值原理和Riccati方程，提出带一步记忆的显式闭环Stackelberg策略，数值实验证明其性能优于反馈策略。
In Stackelberg v/s Stackelberg games a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problems are plagued by the nonuniqueness of follower equilibria and nonconvexity of leader problems whereby the problem of providing sufficient conditions for existence of global or even local equilibria remains largely open. Indeed available existence statements are restrictive and model specific. In this paper, we present what is possibly the first general existence result for equilibria for this c…
In this paper, we investigate a new model of a linear-quadratic mean-field stochastic Stackelberg differential game with one leader and two followers, in which the leader is allowed to stop her strategy at a random time. Our overarching goal is to find the Stackelberg solution of the leader and followers for such a model. By employing the backward induction method, the state equation is divided into two-stage equations. Moreover, by using the maximum principle and the verification theorem, the Stackelberg solution is obtained for such a model.
In this paper, we establish a dynamic game to allocate CSR (Corporate Social Responsibility) to the members of a supply chain. We propose a model of a three-tier supply chain in a decentralized state which includes a supplier, a manufacturer and a retailer. For analyzing supply chain performance in decentralized state and the relationships between the members of the supply chain, we use a Stackelberg game and consider in this paper a hierarchical equilibrium solution for a two-level game. In particular, we formulate a model that crosses through multi-periods with the help of a dynamic discrete…
In this paper, we are concerned with the stabilizatbility of Stackelberg game-based systems. In particular, two players are involved in the system where one is the follower to minimize the related cost function and the other is the leader to stabilize the system. The main contribution is to derive the necessary and sufficient condition for the stabilization of the game-based system. The key technique is to explicitly solve the forward and backward difference equations (FBDEs) based on the maximum principle and give the optimal feedback gain matrix of the leader by using the matrix maximum prin…
This paper studies a class of dynamic Stackelberg games under open-loop information structure with constrained linear agent dynamics and quadratic utility functions. We show two important properties for this class of dynamic Stackelberg games. First, we prove that under mild conditions, the optimal control of individual agents at the solution of the Stackelberg game coincides with the solution to the team problem, where all agents cooperatively achieve the coordinator's objective. Second, we show that the the agent's control at each time step is non-increasing with respect to the coordinator's…
Subsidizing a New Technology: An Impulse Stackelberg Game Approach
针对零排放车辆补贴问题，本文构建含企业连续定价与政府离散补贴干预的脉冲 Stackelberg 博弈，并用验证定理刻画 Feedback Stackelberg 均衡。
Computing Algorithm for an Equilibrium of the Generalized Stackelberg Game
针对1-N广义Stackelberg博弈均衡求解难的问题，提出将其转化为等价1-1博弈并结合投影隐式梯度下降的多项式时间求解方法，通过两个EV充电站定价问题验证了有效性。
Observability Inequality from Measurable Sets and the Stackelberg-Nash Game Problem for Degenerate Parabolic Equations
本研究借助Carleman估计等工具建立退化抛物方程的可测集能观性不等式，应用于Stackelberg-Nash博弈均衡存在性证明，并分析均衡集与范数最优控制问题。
This paper focuses on multi-stage coordination for a population of thermostatically controlled loads (TCL). Each load maximizes the individual utility in response to an energy price, while the coordinator determines the price to maximize the social welfare subject to a peak energy constraint. The coordination problem is formulated as a dynamic Stackelberg game. The Stackelberg solution is derived using an indirect approach: we first obtain an upper bound of the Stackelberg game, then we prove that this upper bound is attainable. Sufficient conditions for the optimal solution are characterized.…
The multilevel reverse Stackelberg game is considered. In this game, the leader controls the outcome by announcing a strategy as a function of decision variables of the followers to his/her own decision space. Corresponding to the leader's strategy, the player in the next level presents his/her strategy as a function of decision variables of the remaining players. This procedure is repeated until it is the turn of the bottom level player in the hierarchy, who reacts by determining his/her optimal decision variables. The structure of this game can be adopted in decentralized multilevel decision…
This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term "overlapping" means that there are common part between the follower's and the leader's information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to represent the state …
A Numerical Analysis for Pursuit-Evasion Games under the Stackelberg Eqilibrium
本文提出了一种在Stackelberg均衡下求解追捕-躲避微分博弈的数值分析框架，引入Semi-DCNLP方法用于飞行路径优化，并验证了该方法的有效性。
The computation of approximate feedback Stackelberg equilibria in multi-player nonlinear constrained dynamic games
本文提出一种高效方法，通过将反馈Stackelberg动态博弈重构为嵌套优化问题并使用牛顿式原始-对偶内点法，计算多人非线性约束动态博弈中的近似局部均衡，并证明了算法的指数收敛性。
The solvability and a Stackelberg-Nash game problem for degenerate elliptic equations
本研究通过加权Sobolev不等式与紧嵌入技术，研究退化椭圆方程弱解空间性质，并应用于证明Stackelberg-Nash博弈均衡的存在性。
This paper considers dynamic (multi-stage) signaling games involving an encoder and a decoder who have subjective models on the cost functions. We consider both Nash (simultaneous-move) and Stackelberg (leader-follower) equilibria of dynamic signaling games under quadratic criteria. For the multi-stage scalar cheap talk, we show that the final stage equilibrium is always quantized and under further conditions the equilibria for all time stages must be quantized. In contrast, the Stackelberg equilibria are always fully revealing. In the multi-stage signaling game where the transmission of a Gau…
This paper is concerned with a three-level stochastic linear-quadratic Stackelberg differential game with asymmetric information, in which three players participate credited as Player 1, Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player 2 acts as the leader of Player 1 and Player 1 acts as the follower. The asymmetric information considered is: the information available to Player 1 is based on the sub-$σ$-algebra of the information available to Player 2, and the information available to Player 2 is based on the sub-$σ$-algebra of the information available to P…
Stochastic Linear-Quadratic Stackelberg Differential Game with Asymmetric Informational Uncertainties: Robust Optimization Approach
本文研究具有非对称信息不确定性的随机线性二次Stackelberg微分博弈，采用鲁棒优化方法，通过求解跟随者的min-max问题和领导者的max-min问题，并利用Riccati方程给出鲁棒Stackelberg均衡的显式状态反馈表示。
We propose a two-layer, semi-decentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establi…
Dynamic Data Pricing: A Mean Field Stackelberg Game Approach
本文提出基于均场Stackelberg博弈的方法，用于动态数据定价问题，通过三级博弈模型分析数据买家、经纪人和卖家之间的战略互动，并推导出近似纳什均衡策略和最优定价策略。
This paper is concerned with the closed-loop solvability of one kind of linear-quadratic Stackelberg stochastic differential game, where the coefficients are deterministic. The notion of the closed-loop solvability is introduced, which require to be independent of the initial state. The follower's problem is solved first, and the closed-loop optimal strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the necessary conditions of the existence of the leader's nonanticipating closed-loop optimal strategy is…
An overlapping information linear-quadratic Stackelberg stochastic differential game with two leaders and two followers
本文研究具有重叠非对称信息的双领导者-双追随者LQ Stackelberg随机微分博弈，通过Riccati方程、最大值原理和CMF-FBSDE方法，获得了Stackelberg-Nash均衡的状态估计反馈表示。
Three-Level Multi-Leader-Follower Incentive Stackelberg Differential Game with $H_\infty$ Constraint
本文研究带$H_\infty$约束的三级多主从激励Stackelberg微分博弈，结合$H_2/H_\infty$控制、凸分析等理论得到三级策略集，实现顶层最优、扰动抑制与跟随者纳什均衡，并用数值算例验证存在性。
Neural Operators Can Play Dynamic Stackelberg Games
本文证明基于注意力的神经算子能近似实现动态 Stackelberg 博弈中跟随者的最佳响应算子，并利用万能逼近定理与博弈稳定性结果证明了该近似博弈的值收敛于原始博弈的值。
Linear-quadratic Stochastic Stackelberg Differential Games with Affine Constraints
本文研究带仿射约束的线性二次随机Stackelberg微分博弈，利用随机Riccati方程和Lagrange对偶理论求解最优策略并给出反馈Stackelberg均衡，同时提出正定条件保证对偶问题解的唯一性。
Open-Loop Stackelberg LQ Difference Games with Coupled-Affine Inequality Constraints: An Exact OCP--LCS/LCQP Reformulation
针对带耦合仿射状态-控制不等式约束的有限时域线性二次Stackelberg差分博弈，提出将其精确重构为最优控制问题与线性互补二次规划的方法，实现该博弈均衡的数值计算并通过约束网络流博弈验证。
Stochastic Mean-Field LQ Stackelberg Differential Games with Random Coefficients: Theory and a Deep FBSDE Picard Solver
本文研究了一种具有随机系数的随机均场线性二次斯塔克尔伯格微分博弈问题，通过扩展拉格朗日乘数法推导出追随者的最优响应的仿射算子表示，并通过深度FBSDE皮卡求解器解决领导者问题，验证了该框架的有效性。
Following the solution to the One-Round Voronoi Game in arXiv:2011.13275, we naturally may want to consider similar games based upon the competitive locating of points and subsequent dividing of territories. In order to appease the tears of White (the first player) after they have potentially been tricked into going first in a game of point-placement, an alternative game (or rather, an extension of the Voronoi game) is the Stackelberg game where all is not lost if Black (the second player) gains over half of the contested area. It turns out that plenty of results can be transferred from One-Ro…
This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$σ$-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate fe…
Stackelberg stopping games
研究离散时间 Dynkin 博弈的 Stackelberg 变体，其中领导者先承诺停止策略，追随者最优响应，导致时间不一致性；通过熵正则化保证随机均衡存在性。
Mean Field Games in a Stackelberg problem with an informed major player
研究知情大玩家向小玩家群体披露私人信息的Stackelberg随机微分博弈，证明其存在松弛解且大玩家的最优控制在有限小玩家博弈中近似最优。
This paper studies a multi-period demand response management problem in the smart grid where multiple utility companies compete among themselves. The user-utility interactions are modeled by a noncooperative game of a Stackelberg type where the interactions among the utility companies are captured through a Nash equilibrium. It is shown that this game has a unique Stackelberg equilibrium at which the utility companies set prices to maximize their revenues (within a Nash game) while the users respond accordingly to maximize their utilities subject to their budget constraints. Closed-form expres…
Mean Field Stackelberg Game for Production and Carbon Emission Reduction with State Reflections
针对带跨界污染与不可逆损害的全球变暖问题，提出含控制量平均场交互与共同噪声的两级斯塔克尔伯格博弈，通过反射机制执行排放上限，证明近似均衡存在性并经蒙特卡洛模拟分析敏感性，为生产与碳减排决策提供方法支持。
This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs), where the coefficients of the backward system and the cost functionals are deterministic, and the control domain is convex. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs is investigated under standard assumptions. The state feedback rep…
Stackelberg Games with a Robust Leader
研究具有一个领导者和多个追随者的斯塔克尔伯格博弈，稳健领导者考虑最坏情况，通过弱形式将问题转化为零和博弈，利用HJB方程刻画，还指出委托代理问题是其特殊情况。
Suboptimal open-loop solution of a Stackelberg linear-quadratic differential game with cheap control of a follower: analytical/numerical study
本文研究了具有小控制成本的Stackelberg线性二次微分博弈的次优开环解，通过分析边值问题的渐进行为，设计了次优控制策略，并通过供应链问题进行示例说明。
In this paper, we consider a discrete-time Stackelberg mean field game with a leader and an infinite number of followers. The leader and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leader commits to a dynamic policy and the followers best respond to that policy and each other. Knowing that the followers would play a mean field game based on her policy, the leader chooses a policy that maximizes her reward. We refer to the resulting outcome as a Stackelberg mean field equilibrium (SMFE). In this paper, we provide a master …
Direct Approach of Linear-Quadratic Stackelberg Mean Field Games of Backward-Forward Stochastic Systems
本文研究涉及后向主导者与大量前向跟随者的线性二次 Stackelberg 平均场博弈，采用直接方法求解，通过平均场近似解耦高维 Hamiltonian 系统，为所有参与者制定分散策略并证明其构成 $(ε_1, ε_2)$-Stackelberg 均衡。
This paper is concerned with a Stackelberg stochastic differential game, where the systems are driven by stochastic differential equation (SDE for short), in which the control enters the randomly disturbed coefficients (drift and diffusion). The control region is postulated to be convex. By making use of the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to establish the Pontryagin's maximum principle for the leader's global Stackelberg solution, within adapted open-loop structure and closed-loop memoryless information one, respectively, w…
Follower Agnostic Methods for Stackelberg Games
提出一种与跟随者无关的高效算法，在领导者未知跟随者效用和策略空间时，利用梯度估计器与适应规则求解多跟随者在线Stackelberg博弈，并证明了其收敛性及在大规模交通网络激励机制设计中的鲁棒性。
Feedback Stackelberg-Nash equilibria in difference games with quasi-hierarchical interactions and inequality constraints
本文针对具有耦合不等式约束的差分博弈，提出反馈Stackelberg-Nash均衡概念，通过动态规划递归求解，并在线性二次情形下转化为线性互补问题，最后用双寡头博弈验证。
Epidemic Control using Stackelberg Mean Field Game with Budget Constraint
本文提出基于预算约束的Stackelberg均值场博弈模型，研究政府如何通过社交距离和疫苗接种策略控制流行病，并给出均衡存在性、唯一性及数值求解算法。
We discuss an open-loop backward Stackelberg differential game involving single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation (BSDE) for which the terminal- instead initial-condition is specified as a priori; the decisions of leader consist of a static terminal-perturbation and a dynamic linear-quadratic control. In addition, the terminal control is subject to (convex-closed) pointwise and (affine) expectation constraints. Both constraints are arising from real applications such as…
Linear-Quadratic Mean Field Stackelberg Stochastic Differential Game with Partial Information and Common Noise
本文研究带部分信息与公共噪声的线性二次平均场Stackelberg随机微分博弈，推导了领导者-跟随者系统的两类分散策略并证明其为ε-Stackelberg-Nash均衡。
A Machine Learning Method for Stackelberg Mean Field Games
针对无法用传统方法求解的 Stackelberg 平均场博弈双层优化问题，提出基于惩罚的单层平均场最优控制重构及神经网络机器学习方法，并证明了其收敛性。
A Partially Observed Stochastic Linear Stackelberg Differential Game with Poisson Jumps under Mean-Variance Criteria
本文提出了一种具有泊松跳跃的随机线性斯塔克尔伯格微分博弈模型，通过正交分解方法解决循环依赖问题，并利用非线性随机过滤技术获得可观察的状态反馈均衡。
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