Non-cooperative Nash Game Research Articles

Stochastic Differential Games of Carbon Emission Reduction in the Four-Tier Supply Chain System Based on Reference Low-Carbon Level

This paper takes corporate social responsibility goodwill and consumers’ reference low-carbon level as endogenous variables of joint carbon emission reduction in the “supplier–manufacturer–retailer–consumer” supply chain system. The joint carbon emission reduction strategies of this four-tier system are analyzed from a dynamic perspective by considering random factors that affect the endogenous variables. Three stochastic differential games are proposed to examine the mechanism between each player, namely the cooperative model, Nash non-cooperative model, and Stackelberg master–slave model. Compared to the Nash non-cooperative game, the manufacturer/supplier-led Stackelberg master–slave game leads to Pareto improvement in the profits of the entire supply chain system and each player. The cooperative game demonstrates the highest expected emission reduction and corporate social responsibility goodwill, but also the highest variance. More importantly, the reference low-carbon level embraces consumers’ subjective initiative in the dynamic of carbon emission reduction. This level is an internal benchmark used to compare against the observed low-carbon level. This paper provides a theoretical foundation for strategic decision-making in emission reduction, contributing to sustainable development. By addressing environmental, economic, and social sustainability, it promotes climate action through carbon reduction strategies and offers policy recommendations aligned with the Sustainable Development Goals.

A two-layer stochastic game approach to reinsurance contracting and competition

We propose a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle. The reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus, and their competition is settled by a noncooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semiclosed form. Furthermore, we obtain interesting sensitivity results for the equilibrium strategies through both analytical and numerical studies.

Strategic coopetition among transportation service providers considering supply–demand congestion effects and asymmetric bargaining power

The phenomenon of transportation service providers (TSPs) engaging in both competition and cooperation, commonly referred to as coopetition, has become increasingly prevalent in the transportation market. This is driven by the rapid advancements in information technology and sharing economy. In practice, competitive TSPs can engage in a resource exchange scheme to share their resources to improve their service quality. However, such a resource exchange scheme may not be sustained since the service prices and profits will be further changed due to the competition in the end market. This study proposes a two-stage sequential-move game to characterize the coopetition problem between TSPs, wherein the first-stage resource exchange problem is modeled with a Nash bargaining game, and the second-stage pricing problem is modeled with a non-cooperative Nash game. Different from prior studies, our model incorporates the supply–demand congestion effects and the asymmetric bargaining power of TSPs. The subsequent impacts on social welfare, TSPs, and end users are investigated. Analytical results show that only when the unit price of the exchanged resources decreases in the exchanged resource quantity will the resource-exchange scheme succeed. Furthermore, we find that TSPs prefer to leave some “buffer zone” in between to avoid fierce competition with price wars.

Knowledge sharing decision-making under stochastic factors in platform ecosystems: the diversified participants' perspective

PurposeKnowledge sharing is critical to creating value in platform ecosystems. However, participants refrain from sharing knowledge and even engage in free-riding behavior, thereby causing the value co-destruction of the platform ecosystems. To encourage knowledge sharing among participants, it is essential to analyze the influencing factors and decision-making mechanisms of knowledge sharing in the platform ecosystems.Design/methodology/approachThe study investigated the issue of knowledge sharing among participants in platform ecosystems, based on the stochastic differential game model. Considering the uncertain factors, the Nash non-cooperative game, Stackelberg leader-follower game, and cooperative game models were proposed. By utilizing system dynamics and numerical simulations, the key influencing factors and mechanisms of knowledge sharing were deeply explored, consequently providing game solutions to achieve the Pareto optimality of the ecosystem.FindingsParticipants' innovation capability and the marginal benefits of knowledge-sharing positively impact knowledge-sharing decisions, while the environmental knowledge decay rate has a negative influence. The platform subsidy mode enhances the knowledge-sharing effect, and the collaborative cooperation mode can realize the Pareto optimization of the system.Practical implicationsThe research findings will provide theoretical support for fostering knowledge innovation and sustainable development of platform ecosystems. Managers should cultivate an innovative environment, establish fair reward mechanisms, and utilize subsidies to promote knowledge sharing, leading to higher value creation.Originality/valueUtilizing the stochastic differential game model, the study proposed various game-theoretic frameworks to analyze participants' knowledge-sharing strategies. The integration of system dynamics and numerical simulations provides a practical approach to understanding the key influencing factors and decision-making processes.

Research on lateral stability control strategy for distributed drive electric vehicles considering driving style

To improve the handling stability of distributed drive electric vehicle (DDEV), a lateral stability control strategy considering driving style is proposed. Firstly, a driving style recognition model based on support vector machines (SVM) was constructed to identify driving styles in different lateral motion scenarios. Then, a layered architecture is used to design a driver/Active Front Steering (AFS)/Direct Yaw Control (DYC) stability coordination control strategy. The upper layer is based on the phase plane of the sideslip angle and extension control theory, dividing the vehicle operating range into classical, extension, and non domains. In the middle level, three control strategies are designed for different work domains. In the classical domain, a driver/AFS coordinated control strategy based on non-cooperative Nash games is constructed, and the weight coefficients in the AFS system cost function are modified based on driving characteristics; Construct a coordinated control strategy for AFS/DYC systems based on cooperative Pareto games in the extension domain; In non-domain, construct a direct yaw torque control strategy based on twin-delayed deep deterministic policy gradient (TD3) reinforcement learning. The lower layer aims to minimize tire utilization and optimizes the allocation of additional yaw moment. Finally, by establishing Hardware in the Loop (HIL) experiment platform, the effectiveness of the proposed control strategy is verified in lane changing scenarios of different styles of drivers. Compared to DYC stability control based solely on TD3, the maximum yaw rate and standard deviation of the lateral stability control strategy considering driving style reduce by 2.4 % and 10.82 %, respectively, and the maximum and standard deviation of the sideslip angle reduce by 29.29 % and 39.24 %, respectively, thereby improving the vehicle's handling stability.

Human–Machine Cooperative Steering Control Based on Non-cooperative Nash Game

Human–Machine Cooperative Steering Control Based on Non-cooperative Nash Game

The Government–Farmer Cooperation Mechanism and Its Implementation Path to Realize the Goals of Optimizing Grain Planting Structure

In order to alleviate the grain supply–demand structural contradictions and ensure the realization of grain planting structure optimization goals, it is necessary to clarify the interactive relationship between multiple entities, establish a cooperation mechanism, and explore its implementation paths. To this end, a differential game model is built to compare and analyze the optimal strategies, optimal benefits, and overall system outcomes for both the government and farmers under three scenarios: the Nash non-cooperative game, the Stackelberg game, and the collaborative cooperation game. Then, key factors and their influencing mechanisms that affect the government–farmer cooperation mechanism are revealed. Finally, the csQCA model is used to explore the implementation paths for different stakeholders to ensure the sound operation of the cooperation mechanism. The results show the following: (1) The government–farmer cooperation mechanism should consist of an inner core system with the government–farmer interaction as the core and an outer system comprising the market environment, cooperation environment, and institutional environment. These two systems should coordinate with each other, respond to each other, and drive progress together. (2) The cooperation mechanism can optimize behavioral enthusiasm, resulting in individual and overall benefits for both the government and farmers. However, its scientific and orderly implementation is affected by factors such as the cost coefficient. Additionally, subsidies serve as a powerful policy tool to enhance farmers’ enthusiasm, thereby increasing the benefits for both parties and maximizing the effectiveness of the cooperation mechanism. (3) There are three implementation paths corresponding to large-scale farmers, rural elites, and small-scale farmers: being led by external policy tools, linkage guidance between decision-making environment and willing subjects, and factor allocation and environmentally driven decision-making. These findings can provide theoretical support and case reference for marginal farmland management and planting structure optimization management in underdeveloped areas.

Stochastic differential game of joint emission reduction in the supply chain based on CSR and carbon cap-and-trade mechanism

This paper considers both carbon emission reduction and corporate social responsibility (CSR) goodwill as endogenous variables in the joint emission reduction process of the supply chain system. The joint emission reduction strategy of the three-tier supply chain system is examined from a long-term perspective utilizing stochastic differential game models. Three models, namely the cooperative game, the Nash non-cooperative game, and the Stackelberg master–slave game, are developed to analyse and compare the equilibrium strategies under each model. The results indicate that the manufacturer-led Stackelberg master–slave game can achieve the Pareto improvement in the profits of the supplier, manufacturer, retailer, and the entire supply chain system.

Design and Pricing of Maintenance Service Contract Based on Nash Non-Cooperative Game Approach

Design and Pricing of Maintenance Service Contract Based on Nash Non-Cooperative Game Approach

A coordinated approach for a three-echelon solar-wind energy supply with government intervention

In the contemporary era, renewable energy sources are employed globally. Their uncertain and intermittent nature conduce to a hybrid system that fully meets the electricity demand. This paper proposes a three-echelon hybrid (solar-wind) supply chain with government intervention and considers two approaches: (1) The decentralized model (non-cooperative Nash game) and (2) The government model with demand-side and supply-side subsidy policies. The optimal values obtained from the Nash game are put in the government function to acquire tariffs, and the paper aims to select the best policy. Finally, sensitivity analysis and managerial insights are presented.

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.

Research on road tracking and anti-roll game control scheme of commercial vehicle based on Nash equilibrium

In view of the problem that the active suspension conflicts with the road tracking system while resisting roll when the commercial vehicle encounters emergency road conditions in the process of unmanned driving, this article is based on the Nash non-cooperative open-loop feedback game theory, taking the road tracking system, and the active anti-roll system as the participants of the game. An interactive control scheme which can not only accurately steer but also take into account the vertical roll stability is proposed. Firstly, the coupled model of Horizontal pendulum-anti-roll for commercial vehicles is established and extended to the vehicle-road three-degree of freedom closed model. Then, based on the derivation of the linear quadratic optimal control model (LQ), the non-cooperative open-loop Nash game theory is used to obtain the interactive optimal control afferent the road tracking system and the active anti-roll system. Finally, the experimental scheme is verified by hardware in the loop experiment. The results show that the interactive control scheme of road tracking and anti-roll based on Nash non-cooperative game for commercial vehicles allows the vehicle to take the vertical stability into account while turning actively, thus ensuring the safety and stability of the vehicle during driving.

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.

Carbon Reduction Subsidy, Remanufacturing Subsidy or Consumer Recycling Subsidy? A Low-Carbon Closed-Loop Supply Chain Network Operation Decision

To promote carbon emission reduction and resource reuse, this study is devoted to studying the impact of government subsidy policies on the operation decision of a low-carbon closed-loop supply chain system. In the production stage, governments can use carbon reduction subsidies and remanufacturing subsidies to reduce the green behavior costs of manufacturers; in the recycling stage, governments can use consumer recycling subsidies to increase the recycling willingness of consumers. In this study, we introduce these three subsidy policies into the target function of the subsidized members. Using the Nash non-cooperative game and variational inequality, we developed a low-carbon supply chain network equilibrium model to examine the impact of subsidy policies on the system operation decision. The results show that, under the three subsidy policies, raising the subsidy level can help to improve the recycling rate, promote the reduction in carbon emissions and improve the profits of retailers; however, manufacturers’ incomes increase only when the carbon reduction subsidy level is below a certain threshold. Notably, under the coexistence of three subsidy policies, the carbon reduction subsidy policy has a more significant impact on the system. Finally, the numerical results show that, when the subsidy level is higher than six, although the increase in carbon reduction subsidy level leads to a decrease in the profits of manufacturers, this policy has the best contribution to the environmental benefits of the system. Our results can serve as guidelines for governments when designing the optimal subsidy programs to achieve the ultimate goal of establishing an environmentally friendly supply chain network system.

Research on open and shared data from government-enterprise cooperation based on a stochastic differential game

<abstract> <p>Based on the perspective of government and enterprises, we explore the cooperative strategy and cost-sharing problem of cooperative open sharing of data between government and enterprises. In order to accurately analyze the data-opening strategies of government and enterprises, stochastic differential game theory is applied to construct the Nash non-cooperative game, Stackelberg master-slave game and cooperative game models with government and enterprises as game subjects to obtain the optimal open data effort, the optimal trajectory of social data open sharing level and the optimal benefit function of government and enterprises in three scenarios. Combined with numerical simulations to analyze the sensitivity of the relevant parameters affecting the level of social data openness, the results of the study revealed the following: ① When the government's income distribution ratio is greater than 1/3, the benefits of the government and the enterprises under the Stackelberg master-slave game and the effort to open and share data are greater than in the Nash non-cooperative situation; in the case of a cooperative game, the degree of effort and total revenue of both parties reach the Pareto optimal state. ② When the government's income distribution ratio is greater than 1/3, the expectation and variance of the open data and shared stock under the cost-sharing situation and the corresponding limit value are all greater than the value in the Nash non-cooperative situation, and in the cooperative game, the expectation and variance of open data and shared stock and its corresponding limit value are the greatest. ③ The government and enterprises coexist with profit and risk under the influence of random interference factors, and high profit means high risk. This research provides a theoretical basis and practical guidance for promoting the open sharing of government and enterprise data.</p> </abstract>

Research on the Coordination of Quality Behavior of Supply 3 Chain of E-Commerce Platform under C2B Model of High-Grade E-Commerce Based on Differential Game

With the increasing demands of consumers for product grade, the C2B model of high-grade e-commerce emerges as required. In order to solve the problem of coordination and cooperation between e-commerce platforms and manufacturers, and to further develop the C2B model of high-grade e-commerce, this paper studies the coordination of supply chain interests by establishing a differential game model considering product grade factors. By comparing the equilibrium solutions of differential games under different decision-making situations, a cooperative coordination mechanism is proposed. Next, the equilibrium solution is further analyzed by means of numerical simulation. Finally, the influence of several important parameters on the equilibrium solution is discussed through sensitivity analysis. It is found that (1) the supply chain parties have optimal quality behavior in the centralized decision-making situation, and the overall benefit is the greatest. (2) Compared with the Nash non-cooperative game, the optimal quality behavior of the dominant party remains unchanged, and the optimal quality behavior of the following party is enhanced after both parties move from the Stackelberg master-slave game, and the optimal profits of both parties, as well as the overall increase. (3) The cooperative coordination model can coordinate the quality behavior of both sides of the supply chain when the benefit distribution coefficient is within a specific range.

Research on the Emission Reduction Decision of Cost-Sharing Logistics Service Supply Chain in the O2O Model

As an effective way to realize energy savings and environmental protection, cost sharing is gradually becoming an important measure to reduce emissions in the logistics service supply chain under O2O mode in recent years. How to conduct contract selection and design optimization under the cost-sharing situation, and then improve the operational efficiency of the logistics service supply chain is an important issue that needs to be addressed. Firstly, based on the initial market demand for logistics, this paper involves the influence of both online logistics service integrators and onsite functional logistics service providers on logistics market demand in terms of emission reduction and platform brand image and develops a model based on the logistics service demand function in the O2O mode. Secondly, for the role of online and onsite emission reduction services in multi-cycle continuous cooperation to enhance the platform integrator’s brand image, a cost-sharing differential game model between online and onsite services is developed to facilitate providers’ adoption of high-quality emission reduction services. Finally, the HJB equation is used to compare the non-cooperative Nash game, the cost-sharing Stackelberg game, and the cooperative game to make the optimal abatement decision, the optimal benefit, and the cost-sharing ratio of the abatement service supply chain in the non-cooperative Nash game, the cost-sharing Stackelberg game, and the cooperative game. By comparing the results of the three games, we find that the optimal onsite and online abatement service decision is related to the cost, marginal revenue, and the impact of the service on demand; the abatement cost-sharing contract and cooperation are both better than the non-cooperative independent decision state, which can effectively guide the provision of high-quality onsite abatement service and improve the revenue of both parties involved in the logistics service supply chain and the total system’s revenue in the O2O mode. Compared with the cooperative game model, the cost-sharing contract can more effectively facilitate close cooperation between the actors, and the relationship between onsite and online marginal revenue affects the improvement of both parties’ revenue. The findings of the study can provide useful managerial insights for the selection and design optimization of abatement contracts for logistics service supply chains considering cost-sharing via the O2O model.

Research on Government-Enterprise Regulation of Online Car-Hailing Based on Differential Game.

In the Internet era, with the widespread application of digital technology, the way people travel has changed. Compared with traditional taxis, more and more people prefer to choose online car-hailing. The rapid development of the online car-hailing industry has solved the problem of taxi-hailing to a certain extent, but it has also brought some new problems. To change the dilemma of the online car-hailing industry, it is necessary to strengthen the regulation of the online car-hailing industry. In this study, we consider the regulatory system composed of a local government and an enterprise and use the differential game to study the regulation of online car-hailing. In the Nash non-cooperative game, Stackelberg master–slave game, and cooperative game, we, respectively, investigate the indicators, such as the optimal regulatory effort of the government, the optimal regulatory effort of the enterprise, the optimal benefit function of the government, the optimal benefit function of the enterprise, the optimal benefit function of the system, the optimal trajectory of the service quality level for the enterprise, and the optimal trajectory of the goodwill for the enterprise. Moreover, we analyze the corresponding conclusions through examples. We obtained some important results. (i) In the Stackelberg master–slave game, the optimal ratio of the local government subsidy to the enterprise's regulatory cost is only related to the benefit distribution coefficient and has nothing to do with other factors. Moreover, when the benefit distribution coefficient is >1/3, the local government is willing to share the regulatory cost of the enterprise. Otherwise, the local government refuses to share the regulatory cost of the enterprise. (ii) Compared with the Nash non-cooperative game, the optimal regulatory effort of the local government remains unchanged in the Stackelberg master–slave game, but the optimal benefit of the local government increases. Moreover, when the benefit distribution coefficient is >1/3, both the optimal regulatory effort and the optimal benefit of the enterprise increase. (iii) Compared with the Stackelberg master–slave game, in the cooperative game, the optimal regulatory effort of both government and enterprise increases, and the system's optimal benefit also increases. (iv) From the Nash non-cooperative game to the Stackelberg master–slave game and then to the cooperative game when the benefit distribution coefficient is >1/3, the service quality level and goodwill of the enterprise all increase.

Prosumer-Driven Voltage Regulation via Coordinated Real and Reactive Power Control

As the penetration of roof-top solar photovoltaics (PV) becomes very large, voltage regulation to the power grid via transactive energy has emerged to achieve system security and flexibility. Several recent studies have extensively studied transactive mechanisms that directly control either real or reactive power of the prosumers. However, they fail to fully incentivize prosumers’ participation due to potential power losses (for real-power control) and prosumers’ low regulation capacities (for reactive-power control). This paper addresses this gap by exploiting the capacities of prosumers’ distributed assets through simultaneous control of both real and reactive powers. In particular, a transactive mechanism is proposed for PV prosumers connected to a remote-area distribution feeder to manage their PV inverters for voltage regulation. In this mechanism, customers are given economic benefits for providing voltage regulation services, which are integrated into their operational objectives. The formulated multi-agent structure is modeled as a non-cooperative Nash game, in which the non-differentiable nature of the problem is eliminated by a smooth approximation, and solved with a Nikaido-Isoda function-based descent direction algorithm. The simulation results show that the proposed scheme incurs lower costs for prosumers both individually and collectively, and significantly improves the feeder voltage profile.

Water pollutant discharge permit allocation based on DEA and non-cooperative game theory

Against the background of the ecological civilization system reform in the new era, the appropriate allocation of water pollutant discharge permits is an important policy for controlling the amount of wastewater discharge. Traditional allocation methods have disadvantages, such as high additional costs, an unfair allocation scheme, and market distortion. In the present study, a fixed-cost allocation model based on data envelopment analysis (DEA) and the Nash non-cooperative game theory is employed to allocate water pollutant discharge permits of totally 31 provinces in China from 2008 to 2017. The allocation scheme considers environmental efficiency. The results demonstrate regional differences in the allocation of water pollutant discharge permits. The eastern region has abundant allocations. The northeastern and central regions have insufficient allocations. Besides, the western region has a significant shortage of allocations. It indicates the higher the utilization efficiency of the water pollutant discharge permits, the higher the region's sustainable development is. Based on the analysis, we propose guidelines for industrial wastewater discharge reduction.