ABSTRACTThis research considers an incomplete information multi‐agent target defense game (TDG) where multiple defenders protect a target against heterogeneous adversaries. The defenders need to distinguish among the adversaries first and then intercept those who pose a threat to the target. Hierarchical reinforcement learning (HRL) approaches provide an efficient solution for multi‐step tasks, and thus show potential in solving the TDG problem. In this article, a novel hierarchical recognition‐allocation‐execution reinforcement learning (HRAE‐RL) approach is proposed. The HRAE‐RL, based on a goal‐directed HRL framework, is composed of a first‐level intention recognizer (IR), a second‐level target allocator (TA), and a third‐level path planner (PP). In IR, an RL‐based compensation structure is proposed to generate stable outputs, which is significant for the lower‐level training, and this structure greatly improves data efficiency. The TA assigns an appropriate adversary as the target of each defender. Differential game theory is utilized as an expert during the training process to predict the optimal interception point, based on which the assignment is obtained. The PP takes advantage of multi‐agent deep deterministic policy gradient to generate cooperative policies for both sides. Simulation results show the superiority and explainability of HRAE‐RL. During execution, the task success rate is not less than 88.47% in various scenarios. In a simple scenario where the optimal solution can be explained analytically, the difference between the average intercept distance generated by HRAE‐RL and the optimal value does not exceed 4.77%.

With the rapid development of metaverse technology, data security governance within its innovation ecosystem has become a critical challenge. This study explores data security collaborative governance decision-making aimed at maximizing energy efficiency and minimizing environmental impact throughout the data lifecycle. It proposes an innovative model for synergistic data security governance and green sustainability, constructing an efficient, secure, and sustainable governance framework. This provides theoretical support and practical guidance for the long-term healthy development of the metaverse ecosystem. Against the backdrop of the metaverse, this paper constructs three game models – Nash non-cooperative, Stackelberg leader-follower, and collaborative cooperative – based on complex systems theory, collaborative governance theory, and differential game theory. From a dynamic perspective, it examines the data security collaborative governance decision-making issues among three key entities: core enterprises, research institutions, and the government. Finally, numerical simulation analysis is conducted. The research findings reveal the following: (1) Government policy support and innovation subsidies can enhance the willingness of core enterprises and research institutions to engage in collaborative governance. Under government incentives and subsidies, the optimal benefits for participating entities and the overall benefits of the ecosystem are improved. (2) The three game mechanisms have heterogeneous effects on improving collaborative governance levels. When the initial level of collaborative governance is low, all three mechanisms can drive its improvement. As the level of collaborative governance increases, the leader-follower game under government incentives promotes better collaborative governance outcomes in the innovation ecosystem. When the level of collaborative governance is very high, only the collaborative cooperation mechanism can further enhance it. (3) Strategies in the cooperative game not only involve optimal decision analysis but also emphasize the promotion of ecosystem integration and optimization through synergistic mechanisms to achieve whole-process, dynamic data security governance, while promoting efficient resource utilization and environmental sustainability, and building a synergistic governance model between data security and green development.

Considering the mutual influence of product prices in adjacent regions on demand, in this paper, we develop a differential game model of transboundary pollution that incorporates emission reduction technologies. By applying optimal control theory, we derive the optimal strategies and pollution stock emission paths for two regions in both non-cooperative and cooperative game scenarios. A comparative analysis of the optimal results and an examination of parameter impacts are conducted for these two scenarios. The research results demonstrate that under cooperative game theory, product prices are lower, optimal production is higher, pollution levels in the air are lower, and social welfare is higher. Furthermore, as price competition intensifies, product prices exhibit a downward trend, while pollution stocks have increased accordingly. Conversely, advancements in emission reduction technology lead to increased regional benefits and a corresponding decrease in pollution stocks.

A Differential Game Method Against DDoS Attacks in IoT Botnets: Holistic and Dynamic Perspectives

PurposeThe purpose of this paper is to explore the enhancement mechanism of quality improvement consulting services in mega water conservancy projects and offer some suggestions to ensure a high level of quality management for such projects which are characterized by their immense scale and lengthy construction periods and exert significant influence on the national economy and social security through their quality.Design/methodology/approachIn this paper, a Stackelberg leader–follower game model based on differential game theory involving four parties—the original supervision party, construction party and owner, along with the newly introduced quality consulting party is constructed. Through simulation and analysis, the effects of incorporating quality improvement consulting services in mega water conservancy projects on eliminating engineering quality issues and changes in the benefits and efforts of each party are analyzed, and the changes in quality management efforts of different parties are obtained.FindingsThe research findings indicate that the involvement of a quality consulting party significantly reduces engineering quality issues; the quality management efforts of both the supervision and construction parties are enhanced, playing a positive role in engineering quality control. Moreover, the introduction of a quality consulting party effectively boosts the project’s capacity to manage quality accident risks. Measures such as increasing the penalty intensity of the project owner and improving the efficiency of quality consulting party's quality management work can effectively elevate the quality management efforts of all parties, further enhancing the level of engineering quality management.Research limitations/implicationsQuality improvement consulting is an effective quality management model and related policies can be formulated to encourage the introduction of quality consulting services, especially in large-scale water conservancy projects. Under its introduction, an increase in the owner’s penalty intensity for quality issues can significantly enhance the quality management efforts of both the supervision party and construction party, providing support for the owner to set appropriate quality control clauses in contracts and strengthen quality supervision.Originality/valueThis study combines Stackelberg leader–follower game theory with differential game theory to create a four-party dynamic game model involving a quality consulting party to analyze its enhancement mechanism. This provides a novel theoretical analysis tool for quality management in mega water conservancy projects. By introducing differential game theory to describe the dynamic process of quality management, this study overcomes the limitation of traditional static analysis, which cannot reflect the impact of the time dimension. The study also proposes a management method including quality consulting party intervention and owner punishment mechanism, offering an innovative approach to implement high-quality, efficient construction of mega water conservancy projects.

Mean-Field Linear-Quadratic Nonzero Sum Stochastic Differential Games with Overlapping Information

In recent years, wetland ecosystems have faced severe degradation, prompting governments to provide carbon compensations to enterprises engaged in wetland conservation efforts. The relationships between governments and enterprises in wetland management are primarily categorized into three models: mission relationships, employment relationships, and alliance relationships. Determining the optimal application scope for each model remains a critical challenge. To address this, this paper constructs three differential game models and conducts a comparative analysis of their equilibrium outcomes. The findings reveal distinct optimal scenarios for governments and enterprises. For governments, the employment relationships model maximizes social benefit when the per-unit benefit of wetland management is small; the mission relationships model is optimal for moderate benefits, and the alliance relationships model for large benefits. For enterprises, the employment relationships model maximizes social benefit when the per-unit benefit is small; the alliance relationships model is optimal for moderate benefits, and the mission relationships model for large benefits.

Key core technology innovation has become an important strategic path for countries to maintain industrial security amid intensifying global technological competition. As an important innovation paradigm, R&D collaboration is generally regarded as an effective way to achieve such innovation. However, the key issue of which collaborative mechanism is most effective at promoting key core technology innovation remains insufficiently explored. Therefore, systematically comparing the effectiveness of different mechanisms of collaborative innovation is of great strategic significance for achieving key core technology innovation and overcoming Western technological blockades. In this study, the R&D level and market share of key core technology were incorporated into an analytical framework and applied to a differential game focused on the innovation behaviors of leading enterprises, supporting enterprises, and academic research institutions under Nash non-collaborative, cost-sharing, and collaborative mechanisms. A simulation analysis was conducted using the MATLAB 2020a software. The results show that the optimal strategies for the key core technology innovation of innovation entities are negatively correlated with the cost coefficient, discount rate, technology, and market recession coefficient. Meanwhile, they are positively correlated with the sensitivity coefficient of technology R&D and market promotion. Furthermore, the R&D levels and market shares of key core technology are highest under the collaborative mechanism. In this scenario, the revenues of the innovation entity and the overall system reach Pareto optimality. Within a threshold range, the cost-sharing mechanism significantly improves innovative efforts, the R&D level, and the market share of key core technology, leading to a Pareto improvement for both the participants’ and overall system’s revenues compared to the non-collaborative mechanism. This study not only contributes to theoretical results of differential games but also provides valuable suggestions for policymakers and innovation entities to foster key core technology innovation from the perspective of collaboration.

ABSTRACTDriven by strategic direction, strategic emerging industry convergence clusters are a complex evolutionary process of increasing technological innovation levels and optimizing industrial structure through cross‐sectoral organizational collaboration and cooperation among various stakeholders. Given the dynamic and long‐term nature of this process, alongside factors such as technological heterogeneity and integration capabilities, we use a differential game approach to compare the optimal innovation strategies across three scenarios: centralized decision‐making, Stackelberg leader–follower, and Nash noncooperative game models. This analysis explores how different innovation entities within strategic emerging industry clusters can coordinate and cooperate to achieve converged cluster development. The results indicate that (1) innovation levels in convergence clusters and the returns of individual actors are lowest under the Nash noncooperative game model, followed by the Stackelberg leader–follower model. The optimal strategy for convergence cluster development is centralized decision‐making and collaborative development. (2) While technological heterogeneity inhibits the benefits of innovation entities, technological integration capabilities increase them. Additionally, the growth of convergence clusters is more strongly impacted by technical heterogeneity, with higher levels of heterogeneity having a negative impact on their development. (3) Under centralized decision‐making, government subsidies have the strongest incentive effect; nevertheless, as compared to other characteristics, their influence on increasing convergence cluster returns is weaker. Findings here may provide theoretical support for enhancing innovation efficiency and promoting strategic emerging industry convergence clusters.

Emergency capability plays a key role in maintaining the stability of data center engineering supply chains. Government subsidies encourage cooperation among supply chain enterprises by offering financial support, aiding in the improvement of emergency preparedness. In this study, the government subsidy rate is treated as an exogenous variable, and a differential game model involving resource suppliers and service operators is constructed. By comparing different cooperation modes, the study investigates the dynamic mechanisms for enhancing emergency capabilities. The conclusions of this study are contingent upon the assumptions of the proposed model. The findings indicate that collaborative cooperation is the most effective approach for improving emergency capabilities and benefits, as it facilitates resource and information sharing. Resource suppliers align their strategies with the overall supply chain, while service operators need to carefully time contracts to optimize cooperation and maximize benefits. Furthermore, government subsidies and the cost coefficient of emergency efforts exert a stronger influence on emergency capabilities than the level of emergency data utilization. To strengthen emergency capabilities, priority should be placed on increasing government subsidies and optimizing emergency effort costs, which can alleviate financial burdens on enterprises and enhance their emergency responsiveness.

PurposeTo incentivize Chinese manufacturing enterprises to increase input in intelligent transformation, the Chinese government typically offers financial and tax incentives as policy support. However, the mechanisms by which these joint tax-subsidy policies affect intelligent transformation input of manufacturing enterprises, and the optimal design of these policies have not been thoroughly explored yet. To address this gap, this study aims to explore the optimal combination of financial subsidies and tax incentives at different stages of the intelligent transformation of manufacturing enterprises in China to maximize the implementation efficiency of financial subsidies and tax incentives.Design/methodology/approachIn this study, a differential game model involving government departments and manufacturing enterprises is constructed by using differential game theory. It explores the boundary conditions and dynamic mechanisms that guide the effective implementation of financial subsidies and tax incentives to foster intelligent transformation. Through simulation analysis, the conclusion is well supported.FindingsFirst, during the demonstration and cultivation stage, financial subsidies are more effective than tax incentives in incentivizing intelligent transformation input of manufacturing enterprises, whereas in the gradual promotion stage, tax incentives are more effective than financial subsidies. Second, the incentive effectiveness of financial subsidies does not change during the process of intelligent transformation, while that of tax incentives increases gradually.Originality/valueThis study fills a research gap by designing the optimal combination of financial subsidies and tax incentives to improve intelligent transformation input of manufacturing enterprises in China. To optimize government resources, during the initial demonstration and cultivation stage, the Chinese government should prioritize high financial subsidies and low tax incentives, while manufacturing enterprises should focus on deploying intelligent transformation projects supported by financial subsidies. In the later stage of gradual promotion, the Chinese government should prioritize high tax incentives and low financial subsidies, while manufacturing enterprises should concentrate on advancing intelligent transformation projects supported by tax incentives.

Min-distance bargaining solution in differential games

Emissions trading scheme's effect on enterprises' sustainable development in China: A differential game and a quasi-natural experiment

Abstract In order to ensure that the load distribution of each distributed power supply is reasonable, we avoid the overload of some distributed power supplies, make full use of all resources, and reduce line losses. A two-layer game operation optimization method for an active distribution network with a high proportion of distributed power is proposed. The output power of the distributed power supply is calculated. The anomaly of the double-layer operation of the active distribution network is identified by the time differential game technique. The optimization index of the two-layer operation stability of the active distribution network is set up. The evaluation function of each optimization index is calculated. The two-layer game operation of an active distribution network with a high proportion of distributed power supply is optimized. The experimental results show that: The output waveform of A phase, B phase, and C phase voltage and current is more stable. The optimization effect is better.

Interpopulation and Intrapopulation Linear Quadratic Differential Games

Dynamic analysis of competitive marketing strategies using differential game models and Runge–Kutta solutions

The growing concern over water scarcity and the management of freshwater is common worldwide. Voluntary incentives, such as payments offered to water users, are recognized as a strategy for reducing conflicts and water consumption. A primary challenge lies in determining the allocation of these incentives among water users, taking into account socio-environmental factors. This study leverages differential game theoretic models to design incentive schemes for the network of water users, geographically distributed across a river network. Two networks of water users are considered: those who utilize groundwater and those who rely on surface water for agricultural purposes. A nongovernmental organization (NGO) participates as another key player, providing conservation incentives to encourage water users to reduce their consumption. The proposed model considers both vertical and horizontal interactions within the network of players. By considering the unique characteristics of each water user, the NGO aims to introduce an incentive scheme designed for each water user. Addressing the challenge of model scalability becomes crucial as the number of players or water users within each network increases, in order to identify optimal decisions. Thus, we propose solution methods for both convex and nonconvex decision problems using Karush-Kuhn-Tucker conditions, Value Iteration, and Basis Function Approximation methods. Finally, we perform parametric analyses to examine how parameters influence the choice of solution methods and affect the decision-making processes. Funding: Financial support from the U.S. National Science Foundation [Grant 2108003] is gratefully acknowledged.

On the solution of Dolichobrachistochrone differential game via dynamic programming approach

To facilitate power system decarbonization, optimizing clean energy integration has emerged as a critical pathway for establishing sustainable power infrastructure. This study addresses the multi-timescale operational challenges inherent in power networks with high renewable penetration, proposing a novel stochastic dynamic programming framework that synergizes intraday microgrid dispatch with a multi-phase carbon cost calculation mechanism. A probabilistic carbon flux quantification model is developed, incorporating source–load carbon flow tracing and nonconvex carbon pricing dynamics to enhance environmental–economic co-optimization constraints. The spatiotemporally coupled multi-microgrid (MMG) coordination paradigm is reformulated as a continuous state-action Markov game process governed by stochastic differential Stackelberg game principles. A communication mechanism-enabled multi-agent twin-delayed deep deterministic policy gradient (CMMA-TD3) algorithm is implemented to achieve Pareto-optimal solutions through cyber–physical collaboration. Results of the measurements in the MMG containing three microgrids show that the proposed approach reduces operation costs by 61.59% and carbon emissions by 27.95% compared to the least effective benchmark solution.

There is a very large literature on applications of stochastic control of jump diffusions and a smaller literature on such games. In many applications it is natural to assume that the arrival intensity is controlled, but except for two long-forgotten papers the literature instead assumes that it is the jump sizes that are controlled. The more natural assumption is typically avoided because a failed Lipschitz condition means that the classical existence and uniqueness proofs cannot be used. We here derive an asymptotic Markov equilibrium of the game with controlled jump intensities and show that it, at least in an example, is very similar to the Markov equilibrium of an analog game with controlled jump sizes. The paper thus makes two contributions: It supplies a way to solve some optimization problems and games with controlled jump intensities and it shows that the commonly used formulation with controlled jump sizes is quite defensible for at least some classes of games.