Stackelberg Differential Game Research Articles

Impacts of dual carbon emission reduction technology and technology spillovers of deterioration reduction on supply chain system's performances considering government incentives and contract design

Investments in carbon emission reduction and deterioration reduction are important aspects for the firms to construct a low-carbon supply chain systems of deteriorating products with the increasing preferences of customers for environmentally friendly and fresh products. In this study we develop a two-echelon supply chain system consists of one manufacturer and one retailer. The manufacturer implements dual carbon emission reduction technology to strengthen the carbon emission reduction efficiency under cap-and-trade regulation and both manufacturer and the retailer invest in deterioration reduction technology to reduce the deterioration and increase the stock level of the fresh items. To increase the demand, the retailer takes the responsibility of promoting the low-carbon and fresh products among the customers. Deterioration reduction technology spills over from the manufacturer to the retailer. Then a differential game model is formulated and the equilibrium strategies are analysed by Stackelberg differential game approach in long term scenario. Numerical analysis has been done to analyse the theoretical results more rigorously by MATLAB software and sensitivity of some major parameters on the demand and profits of the manufacturer and the retailer have also been done to obtain some managerial insights. The impacts of implementing dual carbon emissions reduction technology and the effects technology spillovers are discussed. The results show that when the manufacturer invests in dual carbon emission reduction technology over single reduction technology, it increases the emission reduction level and hence the environmental performance and also the demand and hence economic profits of the both the supply chain members. If the deterioration reduction technology spills over from the manufacturer to retailer, it increases the profit of the manufacturer but the retailer will be beneficial when his/her marginal profit will be high. In addition, we have incorporated government subsidy on manufacturer's emission reduction investment and a cost-sharing contract on the promotional effort of the retailer in our work and then found that joint impacts of them on the supply chain system. We have shown that these two incentives provide the pareto improved solution for the supply chain system and take the supply chain members to a win-win situation as economically and environmentally. The discussions of the results of this study can be beneficial for the industries which produced deteriorating products and emits high carbon during the production process in terms of taking optimal decisions. It also gives a lesson to the government how to inspired the practitioners towards developing a sustainable system.

Two-player nonlinear Stackelberg differential game via off-policy integral reinforcement learning

For the nonzero-sum games, players have taken different strategies to achieve the Nash equilibrium. However, regarding hierarchical optimization and asymmetric information, Nash equilibrium is ineffective. The Stackelberg game provides us with a leader–follower strategy to cope with this problem. This paper solved two-player continuous-time nonlinear Stackelberg differential games with unknown system dynamics using an off-policy integral reinforcement learning (IRL) technique. A two-level hierarchy optimal control problem is the Stackelberg differential game, and locating the open-loop Stackelberg equilibrium is equivalent to solving the coupled Hamiltonian-Jacobi (HJ) equations. First, the optimal strategies for the leader and the follower are derived. Then, the closed-loop system’s asymptotic stability of optimal control solution is proved. Based on the policy iteration (PI) algorithm, an off-policy IRL algorithm is developed to solve the Stackelberg game for completely unknown system dynamics. The control strategies and cost functions of the leader and follower are approximated using neural networks (NNs). It is demonstrated that NNs weight errors of off-policy algorithm are uniformly ultimately bounded (UUB). Lastly, two simulation examples demonstrate that the proposed learning algorithm is capable of obtaining two-player Stackelberg equilibrium strategies.

A Crash Course in Differential Games and Applications

AbstractThe methodology of differential games is a combination of optimal control theory and game theory. It is the natural framework for economic analysis with strategic interaction and dynamical optimization. The theory gained traction by seminal papers in the early seventies, and it gradually found its way into economics. The purpose of this paper is to make theory and applications of differential games easily accessible by explaining the basics and by developing some characteristic applications. The core of the theory focuses on the open-loop and the multiple Markov-perfect Nash equilibria that use the maximum principle and dynamic programming as the techniques to solve the optimal control problems. The applications are the game of international pollution control and the game of managing a lake, which is an example of an ecological system with tipping points. Finally, it is interesting to note that the discovery of time-inconsistency in the open-loop Stackelberg equilibrium had a huge impact on macroeconomics, since policy making under rational expectations is a Stackelberg differential game.

A Stackelberg–Nash equilibrium with investment and reinsurance in mixed leadership game

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization 70(10), 2131–2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results.

The Design of Ecological Compensation for Air Pollution Based on Differential Game

Establishing a scientific ecological compensation mechanism for air pollution is crucial for air protection. This study models the ecological compensation mechanism of the Stackelberg differential game between the local regulator and an enterprise with a competitor by introducing the air quality index and the social welfare benefits of the local regulator. Using the Pontryagin maximum principle, this study obtains dynamic strategies for the local regulator and the enterprise while maximizing the benefits. The evolution of the shadow price is analyzed with the inverse differential equation method. Then, the effects of the shadow price on the optimal dynamic strategies are analyzed using numerical simulation, together with the effects of the introduction of social welfare benefits on the efforts of the local regulator to protect the air environment. The conclusions show that introducing social welfare benefits as an ecological compensation criterion for air pollution promotes air protection by the local regulator.

Zero-Sum Stochastic Linear-Quadratic Stackelberg Differential Games with Jumps

Zero-Sum Stochastic Linear-Quadratic Stackelberg Differential Games with Jumps

Control Allocation of Spacecraft Revisited: A Stackelberg Differential Game Approach

Control Allocation of Spacecraft Revisited: A Stackelberg Differential Game Approach

Robust reinsurance contract and investment with delay under mean-variance framework

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

Reinsurance games with variance-premium reinsurers: from tree to chain

AbstractThis paper studies dynamic reinsurance contracting and competition problems under model ambiguity in a reinsurance market with one primary insurer and n reinsurers, who apply the variance premium principle and who are distinguished by their levels of ambiguity aversion. The insurer negotiates reinsurance policies with all reinsurers simultaneously, which leads to a reinsurance tree structure with full competition among the reinsurers. We model the reinsurance contracting problems between the insurer and reinsurers by Stackelberg differential games and the competition among the reinsurers by a non-cooperative Nash game. We derive equilibrium strategies in semi-closed form for all the companies, whose objective is to maximize their expected surpluses penalized by a squared-error divergence term that measures their ambiguity. We find that, in equilibrium, the insurer purchases a positive amount of proportional reinsurance from each reinsurer. We further show that the insurer always prefers the tree structure to the chain structure, in which the risk of the insurer is shared sequentially among all reinsurers.

Fully-coupled mean-field FBSDE and maximum principle for related optimal control problem

In this paper, motivated by the study of Stackelberg differential game of McKean–Vlasov type, we first investigate the solvability for the mean-field forward–backward stochastic differential equation (MF-FBSDE), which involves one McKean–Vlasov stochastic differential equation (SDE) and one mean-field type backward stochastic differential equation (BSDE). Under some weakly coupled conditions, we use a purely probabilistic approach to prove the well-posedness of such MF-FBSDE with arbitrary fixed time horizon. Then we consider a mean-field control problem, where the state dynamic is given by controlled MF-FBSDE. We establish the stochastic maximum principle and verification theorem for this optimal control problem. We mention that our weakly coupled conditions introduced in studying the solvability for the MF-FBSDE can be applied to simultaneously guarantee the well-posedness of variational equation and adjoint equation which are both fully-coupled MF-FBSDE. As an application, we consider a Stackelberg differential game whose system is described by a McKean–Vlasov SDE. We give a necessary condition for the existence of Stackelberg solutions.

Power Control and Interference Pricing Strategies for Heterogeneous Networks With Renewable Energy Sources

In this paper, the power control and interference pricing problems between the macro base station (MBS) and small cell base stations (SBSs) in Heterogeneous networks (HetNets) are investigated. To optimally allocate power resources and effectively manage the power resources to reduce interference in HetNets, an innovative Stackelberg differential game approach is proposed. In the proposed Stackelberg game, the SBSs control their transmit power to achieve cost minimization based on the interference price offered by the MBS. Given the optimal power control strategies of SBSs, the MBS can adjust the interference price to realize the objective optimization. Renewable energy sources are introduced into the proposed HetNets, and differential equations formulate the SBSs' dynamic energy. Meanwhile, the MBS's state is given by the dynamic spectrum resources. Then the differential game is imported to achieve optimal power control and interference pricing. The open-loop Nash equilibrium solutions are first analyzed for the MBS and SBSs. Then the feedback Nash equilibrium solutions are given. Under the feedback pattern, the proposed model's performances in the non-cooperative and cooperative situations are discussed. Numerical simulations are given to show the convergence and effectiveness of the proposed game approach.

Fuel consumption-reduction investment decisions and coordination contracts in fuel vehicle supply chains: A dynamic analysis

The share of carbon emissions from road transport is as high as 14% of global carbon emissions; fuel vehicles account for a significant part of these emissions. Therefore, there is a long way to reduce fuel consumption in fuel vehicles. We consider the government’s choice of two policies, Vehicle Carbon Emission Regulation (VCER) and Dual Credit Policy (DCP), to regulate the fuel vehicle supply chain and construct a Stackelberg differential game model for fuel consumption-reduction investment decision and operation coordination in a fuel vehicle supply chain consisting of one supplier and one manufacturer. Furthermore, we analyze the supply chain’s equilibrium strategies under centralized and decentralized decision scenarios. Finally, design a bilateral sharing contract to coordinate the supply chain and verify the theoretical analysis by numerical simulation. The results show that the DCP has a stronger incentive for fuel consumption reduction investment than the VCER; the supplier’s profit increases as the severity of the two policies increases, but the change in the manufacturer’s profit is not consistent under the two policies, where the increase in the penalty rate under the VCER decreases the manufacturer’s profit. However, it falls and then rises with tighter fuel consumption standards under the DCP. At the same time, the two-way cost-sharing contracts can encourage more fuel consumption reduction investments while increasing the profits of supply chain members, which is a Pareto improvement for the ecosystem and the supply chain parties.

A stackelberg differential game theoretic approach for analyzing coordination strategies in a supply chain with retailer's premium store brand.

The online version contains supplementary material available at 10.1007/s10479-023-05372-9.

A Stackelberg reinsurance-investment game under α-maxmin mean-variance criterion and stochastic volatility

ABSTRACT This paper investigates a Stackelberg game between an insurer and a reinsurer under the α-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective α-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in [Li, D. & Young, V. R. (2022). Stackelberg differential game for reinsurance: mean-variance framework and random horizon. Insurance: Mathematics and Economics 102, 42–55.], excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity averse or risk averse. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.

A Stackelberg reinsurance-investment game with derivatives trading

This paper studies a stochastic Stackelberg differential reinsurance-investment game with derivatives trading under a stochastic volatility model. The reinsurer who occupies a monopoly position can price a reinsurance premium and invest her wealth in the financial market consisting of a riskless asset and a stock and derivatives tied to the stock. The insurer, the follower of the Stackelberg game, purchases proportional reinsurance from the reinsurer and invests in the same financial market. The main target of the reinsurer and the insurer is to seek their own optimal strategy to maximize the CARA utility of the relative performance. An explicit equilibrium strategy with derivatives trading is deduced by solving Hamilton-Jacobi-Bellman (HJB) equations sequentially. The equilibrium investment strategy demonstrates that the insurer and the reinsurer imitate each other’s investment strategies, showing a herd effect. In numerical experiments, the sensitivity of the equilibrium strategy to model parameters is analyzed. For the optimal investment strategy, we find that a short position in the derivative may switch to a long position with parameters changing, which provides investors with important decision-making information.

Reinsurance games with two reinsurers: Tree versus chain

This paper studies reinsurance contracting and competition in a continuous-time model with ambiguity. The market consists of one insurer and two reinsurers, who apply a generalized expected-value premium principle and a generalized variance premium principle to price reinsurance contracts, respectively. The reinsurance contracting problems between the insurer and reinsurers are resolved by Stackelberg differential games, and the reinsurance competition between two reinsurers is settled by a non-cooperative Nash game. We obtain the closed-form equilibrium strategies for all three players under both a tree structure and a chain structure. A detailed comparison study reveals that the tree structure is preferred to the chain structure from a social planner’s perspective, and the tree structure is generally preferred from the insurer’s perspective.

Investment of Green Technology in a Tourism Supply Chain Under Competition

This article considers a green tourism supply chain (GTSC) composed of one scenic spot (SS) and two travel agencies (TAs) and derives the optimal green technology investments of the SS and TAs in the noncooperation, unilateral cooperation, and bilateral cooperation scenarios by establishing Stackelberg differential games. In addition, we investigate how the green tourism preference of tourists and the competition intensities of the TAs affect the green technology investments of the GTSC members and the greenness levels of tourism products. Our study indicates that the green technology investments of the GTSC members and the greenness levels of tourism products increase with the increasing of tourists’ green tourism preference. We also find that the green technology investments of the TAs and the greenness levels of tourism products change along with the competition intensity of the TAs, but the green technology investment of the SS remains unchanged. Meanwhile, we propose a bilateral cooperation policy that can achieve Pareto improvement of the GTSC performance.

Environmental policies in a Stackelberg differential game

AbstractWe develop a Stackelberg differential game to analyze the economic effects of the reduction plan through two policy instruments, tradable permits and taxes on emissions. Emissions are a by‐product of firm output. The authority acts as a Stackelberg leader, able to set the optimal instrument's level in the light of a finite‐horizon environmental target. We show that the optimal solution of the game is dynamically consistent. Moreover, optimal environmental policies substantially impact the level and composition of economic activity. The differentiation between “clean” and “dirty” firms allows us to assess distributional effects and how environmental technology may influence the game's outcome. Results are shown to be robust under different parameterizations.

Evolutionary Game Theory-Based Optimal Scheduling Strategy for Heterogeneous Computing

With the development of intelligent applications, simply relying on traditional single type of computing unit cannot efficiently satisfy diverse cloud requirements. The emergence of heterogeneous computing can efficiently achieve the adaptation of these intelligent applications by using different types of processing units such as Graphics Processing Unit (GPU) and Field Programmable Gate Array (FPGA). However, the trade-off between profit and costs in the process of scheduling heterogeneous computing resources is also an issue worthy of attention. To address this challenge, this work establishes a heterogeneous computing resource scheduling model based on Stackelberg differential game, which includes three roles Computing Power Trading Platforms (CPTPs), Heterogeneous Computing Service Providers (HCSPs), and Heterogeneous Computing Application Providers (HCAPs). The objective is to maximize utility function of CPTPs and HCSPs subject to rental ratio, pricing strategy and energy consumption of resource scheduling, which has proved that there exists a Stackelberg Nash Equilibrium (NE) solution. The Support Vector Machine based on Artificial Fish (SVM-AF) is proposed to predict the access times of heterogeneous computing applications. In addition, the distributed iteration method and Cauchy distribution is adopted to optimize the computing price strategy and improve its convergence performance. The simulation results show that compared with other strategies, the proposed strategy can effectively improve computing revenue of user experience and while reducing energy consumption in the process of resource scheduling.

A Dynamic Hierarchical Framework for IoT-Assisted Digital Twin Synchronization in the Metaverse

Metaverse, also known as the Internet of 3-D worlds, has recently attracted much attention from both academia and industry. Each virtual subworld, operated by a virtual service provider (VSP), provides a type of virtual service. Digital twins (DTs), namely, digital replicas of physical objects, are key enablers. Generally, a DT belongs to the party that develops it and establishes the communication link between the two worlds. However, in an interoperable metaverse, data-like DTs can be “shared” within the platform. Therefore, one set of DTs can be leveraged by multiple VSPs. As the quality of the shared DTs may not always be satisfying, in this article, we propose an agile solution, i.e., a dynamic hierarchical framework, in which a group of Internet of Things devices in the lower level are incentivized to collectively sense physical objects’ status information and VSPs in the upper level determine synchronization intensities to maximize their payoffs. We adopt an evolutionary game approach to model the devices VSP selections and a simultaneous differential game to model the optimal synchronization intensity control problem. We further extend it as a Stackelberg differential game by considering some VSPs to be first movers. We provide open-loop solutions based on the control theory for both formulations. We theoretically and experimentally show the existence, uniqueness, and stability of the equilibrium to the lower level game and further provide a sensitivity analysis for various system parameters. Experiments show that the proposed dynamic hierarchical game outperforms the baseline.