Differential Game Model Research Articles

Dynamic analysis of dual-market low-carbon supply chain: Considering government intervention and joint promotion

The increasingly severe global environmental problems have urged most people to incorporate environmental protection into their daily lives. Therefore, low-carbon supply chain management has become crucial. This paper studies a dual-market low-carbon supply chain operation in urban and rural markets with one manufacturer and two retailers. Three scenarios are considered: no government intervention, government intervention and joint promotion under government intervention. By establishing and solving three differential game models, we give the dynamic technological innovation, advertising and store promotion strategies of the manufacturer and two retailers and show the impact of government intervention and joint promotion on emissions reduction and performance. The results show that: first, government intervention can increase technology innovation efforts and decrease advertising efforts, while joint store promotion under government intervention has a positive effect on manufacturer's advertising and technological innovation efforts; second, government intervention may increase the profits of the manufacturer, although it will undoubtedly reduce the profits of the retailers, while joint store promotion under government intervention can further improve the profits of the manufacturer and retailers; third, compared to the preferences of rural consumers, manufacturer's technological innovation strategy relies more on the price and promotion preferences of urban consumers. Moreover, the preference differences between the two types of consumers have a weak impact on the manufacturer's advertising strategy, while the price preference of rural consumers has a more significant impact on the advertising efforts of the manufacturer. Overall, the research results can provide valuable suggestions for the members of the low-carbon supply chain in the dual market to improve the economic and environmental performance of the low-carbon supply chain.

Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game

This paper focuses on the saddle point equilibrium strategies for uncertain stochastic hybrid differential game systems under chance theory. We consider systems with dynamics described by both an uncertain differential equation with V-n jumps (UDEV) and a stochastic differential equation (SDE), with linear and quadratic performance index functions. Based on chance theory and the equation of optimality of uncertain stochastic hybrid optimal control model, the saddle equilibrium equation for general uncertain stochastic hybrid differential game system is established. Furthermore, three classes of differential game models are discussed by using equilibrium equation. Finally, an advertising duopoly game problem as an application is provided, the corresponding optimal profit difference value and advertising strategies are analyzed.

Operation Strategies of Green Supply Chain Members with Short-Sighted and Far-Sighted Behavior: A Differential Game Theory Approach

The concept of “green supply chain” has gained increasing attention in the pursuit of sustainable development by enterprises globally, leading to the optimization of supply chain management. However, the behavior patterns of some members in green supply chains exhibit differences. Hence, analyzing the choices of different behavior patterns among green supply chain members is of practical significance in improving supply chain governance and achieving sustainable development. In this study, a differential game model is constructed using differential game theory to analyze the behavior patterns among members of a green supply chain. Analytical solutions under different models are obtained with the aid of Bellman continuous dynamic programming theory, and comparative analysis and numerical examples are carried out to explore the operation strategies of green supply chain members and the choice of short-sighted and far-sighted behavior patterns. Our findings indicate that suppliers playing a leading role in the green supply chain should exhibit far-sighted behavior as decision makers, while retailers should also display far-sighted behavior. This study adopts a dynamic perspective and is innovative and cutting-edge, providing targeted recommendations for enterprises and serving as a reference for the future development of green supply chain management.

Differential game analysis between government and waste incineration plants on the management of municipal solid waste classification

PurposeThe purpose of this paper is to study how the government stimulates incineration plants to participate in waste classification management, and how to adjust the subsidy strategy for them.Design/methodology/approachConsidering that the classification behavior of residents will produce herd effect, and waste classification can reduce the disposal cost of incineration plants, the authors constructed a differential game model between the government and waste incineration plants, and analyzed the input strategy of the government and incineration plants when they cooperate in the management of municipal waste classification.FindingsIncreasing the input level of supervision or raising subsidy price, the government can promote incineration plants to increase the input level of incentive. Moreover, from a long-term perspective, increasing the input level of supervision is more effective. Compared with government supervision, the method of incineration plants incentive can more effectively increase the amount of waste disposal. Furthermore, the government supervision and the incineration plants incentive have a positive interaction effect on improving the amount of waste disposal. Increasing the input level of incineration plants incentive or the level of waste-to-energy technology can increase the amount of waste disposal, and from a long-term perspective, increasing the level of R&D investment is more beneficial to increasing the amount of waste disposal.Originality/valueThe results are helpful to improve the investment in the management of waste classification, and also provide a certain theoretical basis for the government's subsidy policy for incineration plants, so as to reduce the financial pressure of the government.

The optimal carbon emission reduction and advertising strategy with dynamic market share in the supply chain

PurposeAs the quality of the environment decreases, enterprises and consumers' awareness of environmental protection is constantly improving. More and more enterprises begin to increase their investment in carbon emission reduction and attract environmentally friendly consumers to buy low-carbon products through advertising. The purpose of this paper is to utilize a realistic differential game model to provide dynamic carbon emission reduction strategies, advertising strategies and cooperation methods for complex supply chain members from a long-term perspective.Design/methodology/approachThis paper uses the extend Vidale-Wolfe model (V-W model) to discuss the dynamic joint emission reduction strategy in the supply chain.Findings(1) When consumers' awareness of environmental protection increases, on the whole, carbon emission reduction and profit of products show an upward trend. (2) From a long-term perspective, the manufacturer's advertising subsidy to one of the retailers is the best choice. If the strength of the two retailers is unbalanced, the manufacturer will choose to cooperate with the dominant retailer. (3) Advertising, as a marketing means for retailers to promote low-carbon products, can alleviate the adverse effects of prisoner's dilemma in a semi-cooperative state, but it cannot achieve the Pareto optimization result.Research limitations/implicationsThis paper focuses on the analysis of the situation that when the manufacturer is the leader and thinks that consumers are active advocates of low-carbon products.Originality/valueThe results of this paper can provide theoretical basis for the joint emission strategy of supply chain members in low-carbon environment.

Security defense decision method based on potential differential game for complex networks

Most defense strategies in complex networks are developed from the defense perspective, overlooking the key attack-defense characteristics in cybersecurity. A defense decision algorithm is ineffective when dealing with dynamic attacking behaviors, and when based on attack-defense analysis, stochastic uniform network models are generally used to model the target network, while most networks are large and complex. Thus, the algorithms and their results do not well suit small-world, scale-free, and high-aggregation networks. In this study, considering the structural characteristics of complex networks and the attack-defense characteristics of cybersecurity, potential differential game theory is integrated with complex networks, and a global optimal defense decision algorithm is proposed according to the overall network defense objective. Based on the evolutionary analysis of network security states, a network attack-defense potential differential game model is constructed. Adversarial analysis is carried out on the overall attack-defense strategy, and a defense decision algorithm is designed based on a saddle point equilibrium strategy. Simulation tests are carried out on small-world and scale-free networks to evaluate the effectiveness of the proposed method by comparing its performance with that of random defense strategies and classic decision algorithms.

How do university–industry alliances respond to the trust crisis in green technology innovation activities?

PurposeWith the evolution of the turbulent environment constantly triggering the emergence of a trust crisis between organizations, how can university–industry (U–I) alliances respond to the trust crisis when conducting green technology innovation (GTI) activities? This paper aims to address this issue.Design/methodology/approachThe authors examined the process of trust crisis damage, including trust first suffering instantaneous impair as well as subsequently indirectly affecting GTI level, and ultimately hurting the profitability of green innovations. In this paper, a piecewise deterministic dynamic model is deployed to portray the trust and the GTI levels in GTI activities of U–I alliances.FindingsThe authors analyze the equilibrium results under decentralized and centralized decision-making modes to obtain the following conclusions: Trust levels are affected by a combination of hazard and damage (short and long term) rates, shifting from steady growth to decline in the presence of low hazard and damage rates. However, the GTI level has been growing steadily. It is essential to consider factors such as the hazard rate, the damage rate in the short and long terms, and the change in marginal profit in determining whether to pursue an efficiency- or recovery-friendly strategy in the face of a trust crisis. The authors found that two approaches can mitigate trust crisis losses: implementing a centralized decision-making mode (i.e. shared governance) and reducing pre-crisis trust-building investments. This study offers several insights for businesses and academics to respond to a trust crisis.Research limitations/implicationsThe present research can be extended in several directions. Instead of distinguishing attribution of trust crisis, the authors use hazard rate, short- and long-term damage rates and change in marginal profitability to distinguish the scale of trust crises. Future scholars can further add an attribution approach to enrich the classification of trust crises. Moreover, the authors only consider trust crises because of unexpected events in a turbulent environment; in fact, a trust crisis may also be a plateauing process, yet the authors do not study this situation.Practical implicationsFirst, the authors explore what factors affect the level of trust and the level of GTI when a trust crisis occurs. Second, the authors provide guidelines on how businesses and academics can coordinate their trust-building and GTI efforts when faced with a trust crisis in a turbulent environment.Originality/valueFirst, the interaction between psychology and innovation management is explored in this paper. Although empirical studies have shown that trust in U–I alliances is related to innovation performance, and scholars have developed differential game models to portray the GTI process, building a differential game model to explore such an interaction is still scarce. Second, the authors incorporate inter-organizational trust level into the GTI level in university–industry collaboration, applying differential equations to portray the trust building and GTI processes, respectively, to reveal the importance of trust in CTI activities. Third, the authors establish a piecewise deterministic dynamic game model wherein the impact of crisis shocks is not equal to zero, which is inconsistent with most previous studies of Brownian motion.

From depression to wellbeing: How to protect the mental health of isolated people under the "dynamic clearance" of patients with COVID-19.

In 2020, COVID-19 became a global pandemic. The Chinese government's quarantine measures tend to cause anxiety, tension and depression among the quarantined people. This article constructs a differential game model of self-regulation, government guidance and social forces guidance. Then, the psychological benefits of the masses and the benefits of the whole society under the three modes are obtained, and the applicable conditions of various connection modes are compared. The research results show that: compared with social power channeling, the public under the government channeling mode can obtain more psychological benefits. However, with the increase of guidance, the difference between the psychological benefits of different ways of guidance first decreases and then tends to be stable. Under the guidance mode, the social benefits of the government decrease, and the more guidance, the smaller the social benefits. Therefore, both the government and social forces should make use of limited resources to conduct appropriate psychological counseling for the isolated population.

Green Agricultural Products Supply Chain Subsidy Scheme with Green Traceability and Data-Driven Marketing of the Platform

Government subsidies have played an important role in the development of green agriculture. In addition, the Internet platform is becoming a new channel to realize green traceability and promote the sale of agricultural products. In this context, we consider a two-level green agricultural products supply chain (GAPSC) consisting of one supplier and one Internet platform. The supplier makes green R&D investments to produce green agricultural products along with conventional agricultural products, and the platform implements green traceability and data-driven marketing. The differential game models are established under four government subsidy scenarios: no subsidy (NS), consumer subsidy (CS), supplier subsidy (SS), and supplier subsidy with green traceability cost-sharing (TSS). Then, the optimal feedback strategies under each subsidy scenario are derived using Bellman's continuous dynamic programming theory. The comparative static analyses of key parameters are given, and the comparisons among different subsidy scenarios are conducted. Numerical examples are employed to obtain more management insights. The results show that the CS strategy is effective only if the competition intensity between two types of products is below a certain threshold. Compared to the NS scenario, the SS strategy can always improve the supplier's green R&D level, the greenness level, market demand for green agricultural products, and the system's utility. The TSS strategy can build on the SS strategy to further enhance the green traceability level of the platform and the greenness level and demand for green agricultural products due to the advantage of the cost-sharing mechanism. Accordingly, a win-win situation for both parties can be realized under the TSS strategy. However, the positive effect of the cost-sharing mechanism will be weakened as the supplier subsidy increases. Moreover, compared to three other scenarios, the increase in the environmental concern of the platform has a more significant negative impact on the TSS strategy.

Differential Game-Theoretic Models of Cournot Oligopoly with Consideration of the Green Effect

We built and investigated analytically and numerically a differential game model of Cournot oligopoly with consideration of pollution for the general case and the case of symmetrical agents. We conducted a comparative analysis of selfish agents’ behavior (a differential game in normal form), their hierarchical organization (differential Stackelberg games), and cooperation (optimal control problem) using individual and collective indices of relative efficiency. The same analysis wasperformed for the models with the green effect when players chose both output volumes and environmental protection efforts. We used the Pontryagin maximum principle for analytical investigation and the method of qualitatively representative scenarios in simulation modeling for numerical calculations. This method allows for reducing the number of computer simulations, providing sufficient precision. As a result of the comparative analysis, systems of collective and individual preferences were obtained.

A comparative analysis of sustainable exploitation of renewable resources for different information structures

In this paper, a brief description of the authors' concept of sustainable management is given. According to the proposed concept, the sustainable development of an active system means that in the same time some requirements of viability of the system are satisfied, and the interests of the active agents are considered and coordinated. The main emphasis is made on two mathematical examples. First, a relation between egoistic and altruistic evolutionary strategies is studied by means of the Verhulst-Pearl model. Second, selfish and cooperative behavior of the users of a renewable resource is compared on the base of a differential game model. The conditions of viability are formalized as phase constraints. Their satisfaction may be a result of the voluntary coordination of the agents or their enforced coordination provided by a specially introduced principal. The main contribution of the paper consists in the comparative analysis of these ways of sustainable management.

Differential game model of carbon emission reduction decisions with two types of government contracts: Green funding and green technology

In this study, two differential pollution games are constructed according to the impacts of green technology and green funding on promoting carbon emission reduction. In the game, the government provides reasonable green funds or green technologies to encourage enterprises to reduce emissions. Enterprises choose the optimal production plan that maximizes their own interests according to the incentive mechanism of the government. We compare green funding and green technology in terms of total emissions, reduction efficiency and corporate profits. The results demonstrate that: (1) government contracts are not always beneficial for enterprises reducing emissions; (2) when the coverage of green funding is greater than a certain threshold, the emission reduction efficiency brought by green funds is better than that of green technologies; (3) when the funding intensity exceeds a certain value, there exists a optimal coverage of green funding to ensure that corporate profit targets and emission reduction targets are met simultaneously. The results provide a theoretical basis for the government to formulate an emission reduction incentive mechanism.

Motor Vehicle Insurance Anti-Fraud Modeling Based on a Stochastic Differential Game System

In this paper, we regard policyholders, insurance companies, and government departments to be an anti-fraud supervision system, and we explore the supervision of motor vehicle insurance fraud from the perspective of a tripartite game. Taking into consideration the bad reputation records of policyholders as a state variable, through continuous accumulation in effective time, it creates a continuous growth-type warning effect on policyholders, and thus, effectively curbs policyholder fraud and false supervision by insurance companies. At the same time, by considering the influence of random factors on the anti-fraud game of motor vehicle insurance, in this paper, we establish a stochastic differential game model to explore the optimal strategy, the optimal income level, and the expectation and variance of the insured’s bad reputation record stock under the conditions of with and without government supervision. Finally, through a simulation analysis, it is found that the game with government supervision is more conducive to reduce the insured’s fraud intensity, and the simulation proves the impact of different parameters on system stability.

Differential Game Model of Distributed Energy Sharing in Industrial Clusters Based on the Cap-and-Trade Mechanism

The sharing economy is a new economic model that can promote the optimal allocation of resources. Distributed energy sharing in the industrial cluster is of great significance for cluster enterprises to improve energy utilization efficiency and reduce carbon emissions. In this paper, we establish a differential game model for energy sharing in the industrial cluster under the cap-and-trade mechanism analyze the equilibrium strategies of core and supporting enterprises in the industrial cluster under three different decision scenarios. We then conduct a comparative analysis of the results, and the effect of the carbon cap and carbon trading prices on energy sharing in the industrial cluster is discussed in detail. Finally, the results of the theoretical analysis were verified through numerical simulations. The conclusions are as follows: (1) The energy sharing synergy, profit for both parties, and total system profit are the highest under the centralized decision, the Stackelberg game is better than the decentralized decision, and the cost-sharing contract can achieve the overall coordination of interests; (2) A higher carbon trading price can increase the low-carbon level of energy consumption, but interestingly, when the carbon cap is below a certain limit, the increase in carbon trading price within a certain time and interval will lead to a decrease in profits for both parties and total system profit; (3) In the Stackelberg game scenario where cost-sharing contracts are introduced, the cost-sharing ratio of core enterprises will increase as the proportion of benefits to core enterprises increases and government subsidies to supporting enterprises decrease.

Finding an optimal selection in diversify combating modes for desertification region

The differential game method effectively reflects the multi-agent, long-term, and dynamic characteristics of combating regional desertification. Considering the differences in the levels of economic development level and governance technology between adjacent areas, three desertification-combating modes are proposed for exploratory purposes. The differential game models of the two regions combating desertification areas are constructed as per the certain level of cooperation, ecological subsidy, and technology transfer modes. The optimal combating mode is found by comparing the equilibrium results. The study demonstrates that when cooperation level dominated by the region with higher economic and governance technology levels is lower than the threshold $ \phi^{*}_{u_2} $ and $ \phi^{*}_{1V} $, the technology transfer mode is optimal, and the level of cooperation does not affect the technology transfer fee. However, when cooperation level is higher than the threshold $ \phi^{*}_{u_2} $ and $ \phi^{*}_{1V} $, the ecological subsidy mode is optimal. Meanwhile, as the dominant region further raises the cooperation level, it can drive the non-dominant region to increase the desertification combating input level. The certain level of cooperation mode is the least efficient of the three modes. However, when compared with the situation of completely independent governance by the two regions, the certain level of cooperation mode can relatively improve the efficiency of regional desertification combating.

Optimal Strategies Trajectory with Multi-Local-Worlds Graph

This paper constructs a non-cooperative/cooperative stochastic differential game model to prove that the optimal strategies trajectory of agents in a system with a topological configuration of a Multi-Local-World graph would converge into a certain attractor if the system’s configuration is fixed. Due to the economics and management property, almost all systems are divided into several independent Local-Worlds, and the interaction between agents in the system is more complex. The interaction between agents in the same Local-World is defined as a stochastic differential cooperative game; conversely, the interaction between agents in different Local-Worlds is defined as a stochastic differential non-cooperative game. We construct a non-cooperative/cooperative stochastic differential game model to describe the interaction between agents. The solutions of the cooperative and non-cooperative games are obtained by invoking corresponding theories, and then a nonlinear operator is constructed to couple these two solutions together. At last, the optimal strategies trajectory of agents in the system is proven to converge into a certain attractor, which means that strategies trajectory are certainty as time tends to infinity or a large positive integer. It is concluded that the optimal strategy trajectory with a nonlinear operator of cooperative/non-cooperative stochastic differential game between agents can make agents in a certain Local-World coordinate and make the Local-World payment maximize, and can make the all Local-Worlds equilibrated; furthermore, the optimal strategy of the coupled game can converge into a particular attractor that decides the optimal property.

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>

Platform Selling Mode Selection Considering Consumer Reference Effect in Carbon Emission Reduction.

Considering the significant impact of the reference effect on consumer purchasing decisions and corporate profits, this paper mainly focuses on the influence of the reference effect of consumers in carbon emission reduction (CER) on the platform selling mode selection. To this end, this paper establishes a two-level supply chain consisting of a manufacturer who decides on CER in the production process and an online platform that conducts low-carbon publicity. Four differential game models in which the platform uses reselling mode or agency selling mode with or without consumer reference effect are established. The long-term stable cooperation relationship between the manufacturer and the platform, as well as the consumer surplus and social welfare under four models are further investigated. It is found that the reference effect on the platform selling mode is related to the low-carbon publicity effect and commission rate. When the reference effect exists, the intuition indicates that the platform will choose the reselling mode when the commission rate is relatively low. We clarify this result under the condition that the publicity effect is high. However, the manufacturer also prefers platform reselling, which is counterintuitive. When the commission rate is in the middle range, the platform chooses the agency selling mode, which is in line with the preference of the manufacturer. Surprisingly, when the platform's publicity effect is low, the manufacturer and the platform reach stable cooperation in reselling mode when the commission rate is low or high, which is also counterintuitive. When the commission rate is in the middle range, they both prefer the agency selling mode. In addition, it is suggested that the triple benefits in economy, environment, and society are achieved as the optimal selling mode is confirmed in the presence of consumer reference effect in CER.

Pricing and advertising in dual-channel supply chains: real applications for coffee processing and distribution companies

ABSTRACT In the present research, the local and global dynamic advertising and competitive pricing are considered in the form of an integrated differential game model in the dual-channel supply chain of a coffee processing company. Furthermore, integrated mathematical analyses are presented to examine the equilibrium paths of the game. One of the main novelties of this study is to consider the state variable as the amount of credibility that is given to the customer by the global advertising of the manufacturer. According to the results, the equilibrium price of online sales is higher than the retail price. Furthermore, the advertisement volume of the manufacturer in the online channel is more than the local advertisement volume of the retailer, and the advertisement volume of the retailer is more than the global advertisement volume of the manufacturer. In this case, the profit of the manufacturer in the online channel is always more than the profit of the retailer in the traditional retail channel. Therefore, activities of the direct online channel can lead to an increase in profits, however, the conflict of channels could reduce the demands through the traditional retail channel. The sensitivity of equilibrium levels of optimal variables are also analyzed.

Dynamic Pricing and Logistics Service Decisions for Crowd Logistics Platforms with Social Delivery Capacity

With the development of sharing economy, more and more enterprises choose crowd logistics for distribution. Because the crowd logistics platform uses social freelancers, the service quality is difficult to guarantee. Considering the reward-penalty mechanism, dynamic differential game models are constructed to study the optimal pricing and quality of crowd logistics services under stochastic demand based on the optimal control theory and Pontryagin maximum principle. The numerical simulation results show that the optimal dynamic decisions change with the fluctuation of demand dynamically. Furthermore, the platform needs to adjust the value of the reward-penalty factor to ensure the level of service quality and revenue in different situations.