Linear-quadratic Stackelberg Games Research Articles

Deterministic dynamic Stackelberg games: Time-consistent open-loop solution

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: the upper-tier game works according to Stackelberg’s scenario between a leader and a follower, and the outcomes of two lower-tier intertemporal games characterize, respectively, the follower’s and the leader’s response controls that mimic the notion of time-consistent open-loop equilibrium control in existing literature. The obtained open-loop equilibrium of this two-tier game is shown to be weakly time-consistent in the sense that the adopted policies will no longer be denied in the future only if past policies are consistent with the equilibrium policies. On the existence and uniqueness of such a solution, necessary and sufficient conditions are obtained, which are characterized via the solutions of several Riccati-like equations.

A stochastic goodwill model depending on quality level and advertising

This paper focuses on a stochastic goodwill model involving a single manufacturer and a single retailer in a supply chain, in which both quality level and goodwill level are governed by stochastic differential equations. In both the linear-quadratic Stackelberg game and linear-quadratic cooperative game, explicit expressions are obtained for the optimal quality improvement efforts level of the manufacturer, the optimal advertising efforts level of the retailer as well as the optimal profit of the supply chain. We also derive the expectation and variance values for quality and goodwill levels, respectively. The distribution of incremental profit and subsidy are analysed in the cooperative game via the utility theory. Finally, several numerical experiments are reported to show the influence of decay rates and white noise disturbance of the quality and goodwill levels on the expectation of quality and goodwill levels.

Infinite time linear quadratic stackelberg game problem for unknown stochastic discrete‐time systems via adaptive dynamic programming approach

AbstractIn this paper, we propose an adaptive dynamic programming (ADP) approach to solve the infinite horizon linear quadratic (LQ) Stackelberg game problem for unknown stochastic discrete‐time systems with multiple decision makers. Firstly, the stochastic LQ Stackelberg game problem is converted into the deterministic problem by system transformation. Next, a value iteration ADP approach is put forword and the convergence is given. Thirdly, in order to implement the iterative method, back propagation neural network (BPNN) is chosen to design model network, critic network and action network to approximate the unknown systems, objective functions and Stackelberg strategies. Finally, simulation results show that the algorithm is effective.

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.

The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential Games

For stochastic Stackelberg differential games played by a leader and a follower, there are several solution concepts in terms of the players' information sets. In this paper we derive the maximum principle for the leader's global Stackelberg solution under the adapted closed-loop memoryless information structure, where the term global signifies the leader's domination over the entire game duration. As special cases, we study linear quadratic Stackelberg games under both adapted open-loop and adapted closed-loop memoryless information structures, as well as the resulting Riccati equations.

DISCRETE-TIME RICCATI EQUATIONS IN OPEN-LOOP STACKELBERG GAMES WITH TIME PREFERENCE RATES

In this paper the Discrete-Time Riccati Equations in Open-loop Linear-Quadratic Stackelberg Games are studied. In the extended case of games including time preference rates into criteria of the players, it is pointed out that a matrix pencil is characteristic of these Discrete-Time Riccati Equations. A property of the distribution of the characteristic matrix pencil eigenvalues is studied. An approach using deflating subspace of matrix pencil helps to solve the discrete-time Riccati equations. An example illustrates the main result.

Linear feedback closed-loop Stackelberg strategies for descriptor systems with multilevel hierarchy

In this paper, we consider the continuous-time linear-quadratic Stackelberg game for descriptor systems with three levels of hierarchy in the decision-making process. Sufficient conditions for the existence of the linear feedback closed-loop Stackelberg (LFCLS) strategies for both player P1 at the top of the hierarchy and player P2 at the second level of the hierarchy are derived. The obtained LFCLS strategy for P1 forces the other players to act so as to jointly minimize the cost function of P1. The obtained LFCLS strategy for P2 forces player P3 to jointly minimize the cost function of P2 under the declared LFCLS strategy of P1. An illustrative example is included to show the exact existence of such LFCLS strategies.

Closed-loop Stackelberg strategies for singularly perturbed systems: the recursive approach

Abstract The recursive reduced order algorithm in the context of linear closed-loop Stackelberg strategies for singularly perturbed systems is presented. Motivated by previous results for the weakly coupled linear quadratic Nash games, it is shown that a similar algorithm can be produced for singularly perturbed linear quadratic Stackelberg games and the fixed point method is very effective in this case also. It is shown that the proposed recursive algorithm improves the accuracy of the sought solution, that is, the singularly perturbed recursive algorithm converges with the rate of convergence of accuracy O(e). Since all results on linear closed-loop Stackelberg game problem for the singularly perturbed systems can be obtained as far with an accuracy O( e) only, that represents a significant improvement. In order to demonstrate the effectiveness of the proposed algorithm and the failure of the O( e) theory, a numerical example is shown.

Near-Optimal Incentive Stackelberg Strategies for Singularly Perturbed Systems

When the systems contain clow and fast modes. the Stackelberg games are numerical stiffness and ill-conditioning. It is necessary to find some methods to alleviate this ill-conditioning and to reduce the amount of computation. In this paper, we propose a method to treat linear-quadratic Stackelberg games for singularly perturbed systems. To obtain a series of well-conditioned and reduced-order problems, we assume, at first, that the follower responds to the leader’s incentive strategy with composite strategy. A near team-optimal incentive strategy, which contains the follower's composite strategy as information, is constructed by solving these reduced-order problems. And then we prove that the near-team-optimal incentive strategy is well-posed in the sense that the resulting value of the cost function for the leader will have the same limit with the full-order team value of the leader’s cost function as the small singular perturbation parameter e tends to zero, even if the follower responds to the leader’s incentive streategy with full-order strategy.

Closed-form solution for discrete-time linear-quadratic Stackelberg games

A method is proposed for solving the two-point boundary-value problem occurring in discrete-time linear-quadratic Stackelberg games. It is shown that, for open-loop information structure, the necessary conditions can be ordered to form a symplectic matrix. The solution is then obtained by exploiting the properties of such a matrix.

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.