Open-loop Stackelberg Strategy Research Articles

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>

A Payoff Augmentation Approach to Two Pursuers and One Evader Inertial Model Differential Game

In this paper, we study the two-on-one inertial model pursuit evasion differential game with a payoff augmentation approach. In particular, the pursuit evasion game is treated as a deadline game, and its payoff function is augmented to eliminate the deadline as well as to facilitate a more sophisticate model. In addition, we introduce the retrogressive path for analytical solution to the equilibrium strategy for the closed loop game, and proceed to demonstrate that the closed loop game can be converted to a one-side optimization problem for the evader, with the help of the augmented payoff function, under an open loop Stackelberg strategy. More specifically, we establish the conditional equivalency between the open loop solution and the closed loop one. Simulation results verify the effectiveness of the proposed strategy.

A linear quadratic stochastic Stackelberg differential game with time delay

<p style='text-indent:20px;'>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.</p>

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.

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.

A differential game with multi-level of hierarchy

For a multi-level hierarchical differential game governed by a linear equation with quadratic payoffs, the closed-loop representation for the open-loop Stackelberg strategies is derived via a Riccati type equation. Existence and uniqueness of solutions to such a Riccati equation is established.

Stackelberg Solutions for Singular Dynamic Input-Output Models

The first part of the paper is a short review of the dynamic input-output model of macroeconomic systems. By using the input-output approach, a dynamic input-output model with two control variables is presented. With the appropriate performance objective (such as the consumption-tracking control criterion), a dynamic game model of the macroeconomic system is constructed. In the second part, the Stackelberg strategy concept is introduced to the dynamic game given in the first part. By using the standard variational techniques, the necessary conditions for the existence of open-loop Stackelberg strategy is derived, and the open-loop Stackelberg strategy is given. In addition, an example is provided for illustrating the results of the paper.

On structural control strategies in aquatic ecosystems

Three different control strategies are designed for a class of structural nonlinear predator-prey models (nutrient-herbivore-predator) with natural and man's control separated. The optimal open-loop control, the open-loop Stackelberg strategy and two-level control strategy which is based on the coordination in the Stackelberg sense, and the local multiple predator-prey models control employing the network flow strategy are described. The sampled data systems are considered to compared these strategies.

Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay

Abstract The open-loop Stackelberg strategies in a linear-quadratic differential game with time delay are constructed in the form of necessary and sufficient conditions. The evolution of a game with time delay is described by coupled differential equations composed of lumped and distributed parameter subsystems. Calculation of solutions is facilitated by transforming two-point boundary-value problems to terminal boundary-value ones by the introduction of Riccati-type differential equations.

Solution of N-person Stackelberg games with nearly cooperating leaders

A class of linear-quadratic Stackelberg games with many leaders and many followers is considered. For this game, a proportionality relation is assumed between some of the weighting matrices in the leaders’ cost functions. With this assumption, it is shown that the matrix characterizing the set of necessary conditions to be satisfied by an open-loop Stackelberg strategy has a special spectrum. This property is then used to solve the two-point boundary-value problem (TPBVP) associated with the game by an eigenvector method.

Analytical solution for an open-loop Stackelberg game

A method to solve the two point boundary value problem (TPBVP) occurring in open-loop Stackelberg strategy for linear-quadratic games is presented. By writing in a special order the necessary conditions that an open-loop Stackelberg solution must satisfy, a Hamiltonian matrix comes out and its properties are exploited to solve the TPBVP. The solution procedure becomes similar to the one used to solve a matrix Riccati equation by the eigenvector method. A simple example is given to illustrate the proposed method.

Stackelberg strategy for linear quadratic partial differential games

Characterizations of open-loop Stackelberg strategy for two-person nonzero-sum dynamic games described by linear operators (including unbounded ones) in Hilbert spaces are presented. The method of Lions is used and it guarantees the existence, uniqueness, as well as necessary and sufficient conditions for the open-loop Stackelberg strategy of such games. Special cases considered include games governed by ordinary differential equations, linear elliptic and parabolic partial differential equations with quadratic cost functions.

Stackelburg solution for two-person games with biased information patterns

A strategy suggested by Stackelberg for static economic competition is considered and extended to the case of dynamic games with biased information pattern. This strategy is reasonable when one of the players knows only his own cost function but the other player knows both cost functions. As with Nash strategies for nonzero-sum dynamic games open-loop and feedback Stackelberg strategies for dynamic games could lead to different solutions, a phenomenon which does not occur in optimum control problems. Necessary conditions for open-loop Stackelberg strategies are presented. Dynamic programming is used to define feedback Stackelberg strategies for discrete-time games. A simple resource allocation example illustrates the solution concept.

Matrix Quadratic Solutions

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients