Three-level multi-leader-follower incentive Stackelberg differential game with $H_\infty$ constraint

In this paper, an infinite-horizon incentive Stackelberg game with multiple leaders and multiple followers is investigated for a class of linear stochastic systems with H ∞ constraint. In this game, an incentive structure is developed in such a way that leaders achieve Nash equilibrium attenuating the disturbance under H ∞ constraint. Simultaneously, followers achieve their Nash equilibrium ensuring the incentive Stackelberg strategies of the leaders while the worst-case disturbance is considered. In our research, it is shown that by solving some cross-coupled stochastic algebraic Riccati equations (CCSAREs) and matrix algebraic equations (MAEs) the incentive Stackelberg strategy set can be obtained. Finally, to demonstrate the effectiveness of our proposed scheme, a numerical example is solved.

This paper is mainly concerned with the derivation of the sufficient conditions for the incentive Stackelberg strategy in the two-level hierarchical differential games with two noncooperative followers, characterized by a classof linear state dynamics and quadratic cost functionals. In the paper, we first give some concepts in the two-level hierarchical games with two noncooperative followers, and by a simple numerical example, show a general method of solving such a two-level incentive static game problem. Then, we construct a new form of the incentive Stackelberg strategy \(\bar \gamma _0 = \left\{ {\bar \gamma _{0 1} ,\bar \gamma _{0 2} } \right\}\) of the leader P0, and also derive the sufficient conditions which are satisfied by this strategy \(\bar \gamma _0 = \left\{ {\bar \gamma _{0 1} ,\bar \gamma _{0 2} } \right\}\).

In this paper, an open-loop two-person non-zero sum stochastic differential game is investigated for fully coupled forward-backward stochastic system driven by a Brownian motion and a Poisson random measure.A variational formula for the cost functionals is obtained directly in terms of the Hamiltonian and the associated adjoint system which is a linear FBSDEs and neither the variational systems nor the corresponding Taylor type expansions of the state process and the cost functional will be considered. As an application, one necessary condition (a stochastic maximum principle) and one sufficient condition (a verification theorem) for the existence of open-loop Nash equilibrium points are proved by the variation formula obtained in a unified way. The control domain need to be convex and the admissible controls for both players are allowed to appear in both the drift and diffusion of the state equations.

This letter investigates an infinite-horizon incentive Stackelberg game for discrete-time linear stochastic systems subject to Markov jump parameters and external disturbance by means of static output-feedback (SOF). In contrast to the existing studies, players can only have access to local output information in designing their incentive Stackelberg strategies with H ∞ constraint. It is shown that the incentive Stackelberg strategy set is obtained by solving a set of higher-order cross-coupled Lyapunov type equations. As another important contribution, a numerical algorithm is proposed based on the coordinate descent method that guarantees local convergence. A numerical example demonstrates the existence of the SOF incentive Stackelberg strategy set and the effectiveness of the proposed algorithm.

This technical note is concerned with the maximum principle for a non-zero sum stochastic differential game with discrete and distributed delays. Not only the state variable, but also control variables of players involve discrete and distributed delays. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, the author establishes a necessary maximum principle and a sufficient verification theorem. To explain theoretical results, the author applies them to a dynamic advertising game problem.

In this paper, incentive Stackelberg games are investigated for a class of stochastic linear systems. Unlike the existing ordinary Stackelberg games, output feedback control is developed in such a way that state variables cannot be measured directly, especially for networked control systems. First, the incentive Stackelberg strategy is designed by the leader to achieve team-optimal solution. Second, the follower achieves its equilibrium ensuring the incentive Stackelberg strategy by solving some cross-coupled stochastic Riccati differential equations. Finally, the optimal estimator (conditional expectation) for the leader and follower is derived.

This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the gamesystems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls.Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibriumpoint of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investmentportfolio and optimal impulse consumption strategy are obtained explicitly.

We prove the existence of a value for pursuit games with state constraints. We also prove that this value is lower semicontinuous.

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.

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.

This paper deal with a kind of traffic pricing control problems of communication networks. The incentive Stackelberg strategy concept in the game theory was introduced to the network traffic model that comprises subsidiary systems of users and network. A linear incentive strategy, a nonlinear incentive strategy and a self-reported traffic modle with linear incentive strategy were proposed to the traffic problem. Some numerical examples and simulations were given to illustrate the proposed method.

Dynamic games with hierarchical structure have been identified as key components of modern control systems that enable the integration of renewable cooperative and/or non-cooperative control such as distributed multi-agent systems. Although the incentive Stackelberg strategy has been admitted as the hierarchical strategy that induces the behavior of the decision maker as that of the follower, the followers optimize their costs under incentives without a specific information. Therefore, leaders succeed in using the required strategy to induce the behavior of their followers. This concept is considered very useful and reliable in some practical cases. In this survey, incentive Stackelberg games for deterministic and stochastic linear systems with external disturbance are addressed. The induced features of the hierarchical strategy in the considered models, including stochastic systems governed by Ito stochastic differential equation, Markov jump linear systems, and linear parameter varying (LPV) systems, are explained in detail. Furthermore, basic concepts based on the H2∕H∞ control setting for the incentive Stackelberg games are reviewed. Next, it is shown that the required set of strategies can be designed by solving higher-order cross-coupled algebraic Riccati-type equations. Finally, as a partial roadmap for the development of the underdeveloped pieces, some open problems are introduced.

We develop a new constructive method for proving the existence of Nash equilibrium for a class of nonzero sum stochastic differential games. Under certain usual assumptions, we prove the existence of Nash equilibrium for discounted payoff criteria. A novel feature of our method is that it allows us to compute Nash equilibrium for a large class of stochastic differential games.

In this article, we study a class of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding Nash equilibrium control is required to be adapted to the filtration generated by the observation process. To find the Nash equilibrium point, we establish the maximum principle as a necessary condition and derive the verification theorem as a sufficient condition. Applying the theoretical results and stochastic filtering theory, we obtain the explicit investment strategy of a partial information financial problem.

This technical note is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by backward stochastic differential equations (BSDEs). This kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities. We establish a necessary condition and a sufficient condition in the form of maximum principle for open-loop equilibrium point of the foregoing games respectively. To explain the theoretical results, we use them to study a financial problem.