The Carbon Border Adjustment Mechanism (CBAM) is a border tax implemented by highly carbon-regulation countries (HCRC) on imports from less carbon-regulation countries (LCRC). To analyse carbon reduction cooperation in transnational supply chains under CBAM, this study develops a biform differential game model involving a distributor in HCRC and two manufacturers (from HCRC and LCRC) supplying homogeneous products. Within this framework, supply chain members cooperate in carbon reduction while independently setting the wholesale prices or retail prices. The findings reveal a key distinction between cooperative and non-cooperative modes: under non-cooperation, pricing and emission reduction decisions remain static, whereas under cooperation, they dynamically evolve and gradually stabilise. Moreover, as the CBAM price increases, distributors in the cooperative model increase the LCRC product sales but reduce them in the non-cooperative setting. Consequently, under cooperation, the distributor’s profit increases with CBAM prices, while under non-cooperation, the profit follows an inverse U-shaped pattern. Finally, the cooperative mechanism enhances sales, emission reductions, and both LCRC and HCRC manufacturers’ profits compared to the non-cooperative model. However, the distributor achieves higher profits only when CBAM prices are sufficiently high, and thus should adopt the cooperative strategy under the same condition.

Abstract The development of Geographical Indication Agricultural Products (GIAP) plays a critical role in promoting rural revitalization. However, due to information asymmetry among stakeholders, producers often tend to pursue opportunistic gains by “free‐riding,” such as producing low‐quality agricultural products, which severely damages the reputation of GIAP. By improving information provision, digital technologies offer a means to eliminate this information asymmetry, facilitate stakeholders' sharing of GIAP production and quality information, and ultimately make GIAP governance possible. Based on this, the paper first discusses the digital information sharing mechanisms of GIAP under different collaboration models. Next, we develop a differential game model based on information sharing. Finally, we compare the differences in information sharing efforts and benefits among various stakeholders. Results show that (i) under the context of digital technology, GIAP's information‐sharing behaviors can be categorized into three modes: decentralized, subsidy, and centralized; (ii) as the cooperation model progresses, the information‐sharing efforts of the government, industry associations, and enterprises increase to varying degrees within a specific range of coefficients; (iii) in the enhancement pathways for information sharing among the three parties, there exists a Pareto improvement path that yields benefits; and (iv) the “consortium” model under centralized decision‐making can achieve dual maximization of information sharing efforts and benefits for all parties involved through the Pareto improvement path.

This study addresses the bilateral capital constraints for research and development (R&D) and blockchain (BC) investments in the new energy vehicles (NEV) sector. By employing differential game theory and continuous dynamic programming, it explores the pricing and technology investment strategies of NEV company under different financing portfolio modes and reveals the dynamic evolution of corporate goodwill and supply chain performance. From the perspectives of initial capital and financing interest rates, the study identifies the ‘bank loan + deferred payment’ mode as optimal. Leveraging its capital suitability and synergy between technology investment and pricing, this mode effectively promotes the simultaneous growth of supply chain goodwill and demand. By incorporating demand uncertainty, the study demonstrates that risk premiums induced by stochastic demand and risk aversion negatively impact the supply chain. Innovatively, a synergistic mechanism integrating bidirectional cost-sharing contract with financing is proposed. This mechanism not only incentivizes companies to increase technological investment but also balances consumer utility compensation, thereby establishing a virtuous cycle of optimized pricing and enhanced goodwill, ultimately achieving superior overall benefits compared to an unconstrained supply chain. The findings provide theoretical support and practical pathways for NEV companies in formulating technology innovation investment and dynamic optimization strategies.

Interaction between the Government and the Central Bank to Stabilize Public Debt: A Differential Game Approach

This paper investigates the turnpike properties of deterministic nonzero-sum linear-quadratic (LQ) differential games. Under certain assumptions on the Hamiltonian matrix of the nonzero-sum LQ differential game, we establish the solvability of both the coupled non-symmetric differential Riccati equation (DRE) and the algebraic Riccati equation (ARE). Moreover, we identify the convergence relationship between the DRE and ARE, which is essential for understanding the turnpike properties. Over a finite but sufficiently long time horizon, the open-loop Nash equilibrium is shown to remain exponentially close to the solution of a two-objective optimization problem for the majority of the time horizon.

Research on marine accident rescue based on differential games

Differential Graphical Game for Linear Multi-Agent Systems With Deceptive Agents or Traitorous Agents

Maximum principle for partial information non-zero sum stochastic differential games with mixed delays

Strategy fusion and differential game based guidance law for multi-missile with collision avoidance

Iterative Stackelberg equilibrium finding for linear quadratic differential games

Zero-Sum Differential Game Based Fault-Tolerant Control for a Class of Nonlinear Multi-Agent Systems With Communication Faults

Multi-pursuer single-evader privacy-preserving differential games

Based on differential game theory, this article studies the competition and cooperation in the supply chain system of the new energy vehicle industry. It constructs profit functions for new energy vehicle manufacturers and core component suppliers in two different situations: competitive decision-making and cooperative decision-making. The optimal factor values and profit values of each subject are obtained, and case analysis is conducted. Research has found that: (1) Whether in competitive or cooperative decision-making situations, the optimal factor values of each subject are influenced by their respective costs and show an increasing trend. (2) By comparison, it is found that the optimal profit of new energy vehicle manufacturers under cooperative decision-making and the overall optimal profit of the new energy vehicle industry supply chain system are better than those under competitive decision-making. (3) The profit growth rate of core component suppliers of new energy vehicles and the supply chain system of the new energy vehicle industry under cooperative decision-making is significantly faster than that under competitive decision-making.

To effectively balance emission reduction and preservation in the agricultural product sales system, this study models the carbon emission allocation of the agricultural product supply chain based on differential game theory. Starting from decentralized and centralized strategies without subsidies, research and modeling are conducted on the influencing factors of emission reduction, publicity, and preservation. Secondly, considering the scenario of government subsidies, the research analyzes the dynamic impact of different emission reduction, publicity, and preservation measures on the operation of the system. In the unsubsidized analysis model, the trajectory of goodwill shows variability, while the trajectory of freshness and emission reduction is mainly monotonic. In addition, freshness preference can enhance emission reduction decisions, and centralized strategies are superior to decentralized strategies in emission reduction. The increase in goodwill preference enhances the preservation, emission reduction, and sales status variables of agricultural products, but the centralized strategy is better than the decentralized strategy. Under the bilateral coordination mechanism, the maximum profit of the centralized strategy and bidirectional cost coordination mechanism system is 4350. In addition, under the two-way cost coordination mechanism, both retailers and suppliers have the highest profits, which are 1950 and 2352 respectively. In subsidy analysis, when the profit ratio is 0.5, the benefits of retailers and sellers reach equilibrium. In addition, bilateral coordination mechanisms have better system economic benefits, environmental benefits, and social welfare. This study is beneficial for improving carbon emissions from agricultural products and enhancing the effectiveness of agricultural product market development. This study provides technical support for agricultural emissions reduction and optimization of agricultural product supply chains.

Retraction notice to "Differential game model and coordination model for green supply chain based on green technology research and development" [Heliyon 7 (2021) e07811

Under the dual carbon target regulation, this article constructs a collaborative research and development carbon reduction model for the photovoltaic industry supply chain from the perspective of carbon tax and consumer green preferences, using differential game theory. Considering three different scenarios of no research and development carbon reduction, independent research and development carbon reduction, and collaborative research and development carbon reduction, the optimal factors and profit values are obtained, and case analysis and sensitivity analysis are conducted. Research has found that: 1) The optimal carbon reduction achieved by photovoltaic industry supply chain entities through cooperative research and development is higher than that achieved through independent research and development. 2) The increase in carbon tax rates has to some extent increased carbon emissions, but at the same time reduced the overall profit of the photovoltaic industry chain. 3) The higher the proportion of research and development costs borne by photovoltaic system manufacturers, the higher the carbon emission reduction of photovoltaic silicon wafer suppliers' research and development. 4) Consumer green preferences are beneficial for increasing carbon emissions reduction in the photovoltaic industry chain.

Stochastic differential games for optimal investment and risk control problems in an incomplete market

In the process of studying mathematics, it is important to show the application of its results in other disciplines. The use of mathematics in the study of special disciplines can be carried out in the process of interdisciplinary connections, as well as in solving applied problems. In various applications, particularly in analytical mechanics, it is possible to use certain provisions of the theory of differential games, namely, the conditions for the existence of equilibrium situations in non-cooperative differential games of several persons. This article presents studies of the necessary conditions for the existence of an equilibrium situation, using some concepts and principles of analytical mechanics. Thus, to define the action according to Hamilton, the necessary conditions were obtained in the form of the Euler-Lagrange equations. This form of necessary conditions in N-person differential games is of interest to students of natural sciences.

A two-player differential game with an unfixed endpoint is considered. A special feature of the game is the presence of not only a target set but also a lifeline. If the second player steers the lifeline, then the payoff equals infinity. The payoff functional depends on the trajectory of the players and their controls. Special cases of the differential game under consideration are the pursuit--evasion game and time-optimal game. Universal positional strategies are constructed for the game under consideration under the assumption that the Dirichlet problem for the Hamilton--Jacobi equation, related to the differential game, admits a viscosity proximal solution. The construction of universal strategies is based on the concept of a proximal gradient and utilizes the Krasovsky--Subbotin approach. The universality of positional strategies means that for any initial point from a compact set, the feedback strategy is equally effective. In addition, theorems on the evaluation of the guaranteed result of the players are proved.

Introduction Food safety incidents in the food processing industry have frequently occurred, exposing significant management loopholes in the food supply chain. To address these deficiencies, digital traceability serves as a promising solution. However, its implementation alters profit structures, creating new supply chain coordination challenges. Methods To investigate these changes, we develop a differential game model between a supplier and a processor. This study aims to systematically analyze the influence of digital traceability on food safety and supply chain profits. On this basis, it further explores effective methods for profit distribution and the key conditions required for their success. Results The research results are as follows. First, digital traceability is effective in improving the safety of food raw materials and food, except in cases where profit is distributed based on safety level or supplier effort, combined with an insufficient profit share for the supplier. Second, when the raw material safety attenuation factor is high, it is more favorable to distribute profit based on the raw material safety level, and when it is low, it is more favorable to distribute profit based on the supplier's effort. Third, in the initial distribution modes, digital traceability usually leads to a decrease in supplier profit and an increase in processor profit. In other modes, the effect depends on the distribution coefficient. Discussion Therefore, the selection of an appropriate profit distribution plan must be determined by key parameters such as the decay rate of raw material safety levels and the penalty amount, to ensure both food safety and mutual profitability.