A Partially Observed Nonzero-Sum Differential Game of Mean-Field Backward Doubly Stochastic Systems

Abstract In this paper, we investigate a competitive market involving two agents who consider both their own wealth and the wealth gap with their opponent. Both agents can invest in a financial market consisting of a risk-free asset and a risky asset, under conditions where model parameters are partially or completely unknown. This setup gives rise to a nonzero-sum differential game within the framework of reinforcement learning (RL). Each agent aims to maximize his own Choquet-regularized, time-inconsistent mean-variance objective. Adopting the dynamic programming approach, we derive a time-consistent Nash equilibrium strategy in a general incomplete market setting. Under the additional assumption of a Gaussian mean return model, we obtain an explicit analytical solution, which facilitates the development of a practical RL algorithm. Notably, the proposed algorithm achieves uniform convergence, even though the conventional policy improvement theorem does not apply to the equilibrium policy. Numerical experiments demonstrate the robustness and effectiveness of the algorithm, underscoring its potential for practical implementation.

Background As destructive diseases, governments increasingly utilize emergency co-production to satisfy the keen need for medical products. Although governments have provided purchase commitment payments to promote co-production with manufacturers, research has mainly focused on regular medical production without purchase commitment payment, whereas research on emergency co-production with purchase commitments remains scarce. Method Our study fills the gap by analyzing the regular and emergency co-production models using differential game approaches to optimize purchase commitment. Results First, we find that even if the production and storage costs are relatively high, the emergency co-production mode sometimes has a higher production rate than the regular medical production mode. Next, the purchase commitment payment could increase the relative advantage of emergency co-production in the per-unit product value. Moreover, the emergency co-production model dominates when the demand exceeds a threshold. Furthermore, the regular medical production mode dominates the higher value and less-demanding emergency production. Conclusions Hence, both regular medical production and co-production modes can show superior performance depending on the level of demand and the degree of purchase commitment payment. Critical management insights offer takeaways for manufacturers and policy makers' decisions on emergency medical co-production to prepare for future onslaughts of destructive diseases.

Agent or merchant? Unpacking hotel-OTA selling formats with differential games

Integrating opinion dynamics and differential game modeling for sustainable groundwater management.

BackgroundThe new media environment driven by digital intelligence technologies provides new opportunities for enterprise development, but also brings new challenges for enterprise crisis management and network public opinion information (referred as public opinion) guidance. Focusing on the contradictions between the complex diversity of crisis types and the limited governance resources, it is of great significance for enterprises to determine the best public opinion guidance strategy.MethodsConsidering the dynamic complexity of public opinion derived from enterprise crisis, this paper innovatively proposed a decision-making model of enterprise public opinion under crisis differentiation based on differential game. Then, the balance strategy of each stakeholder and the guiding effect of public opinion were discussed under four decision scenarios. Finally, the optimal resources allocation ratio of different types of crises is determined under the constraints of governance resources, and the key parameters of public opinion guidance process are identified.ResultsThe results show that, under four decision scenarios, the dual-guidance strategy achieves the best public opinion guidance effect and Pareto-optimal outcome for the game system. The overall benefit is maximized when the optimal investment ratio for the four types of crises (Values-type, Product-type, Marketing-type, and Internal management-type) is 60:28:8.4:3.6. The subsidy coefficients of enterprise to netizens and media both significantly the guidance effect of public opinion, but compared with the former, it is more sensitive to the changes of the letter.ConclusionsBased on the research results, this paper provides targeted suggestions for enterprises crisis response and their sustainable development under the new media environment.

Taking advantage of high-performance intelligent robots to solve the coordination control problem such as assembly, handling, and installation, transportation is gradually becoming a kind of frontier subject with great scientific research value in the field of robotics. However, due to possible conflicts and inconsistencies between the manipulator and the operating object, it is challenging to design the optimal coordination control scheme between human and robot. This article presents an event-triggered mixed nonzero-sum game optimal control method, which considers both nonzero-sum game and cooperative game cases, for modular robotic manipulator (MRM) systems performing coordinated operation tasks. First, the joint torque feedback technique and joint task assignment method are employed to establish the dynamic model of MRM subsystem, and then, the global state-space description is deduced. For the unknown information containing interconnected dynamic coupling (IDC) terms and friction modeling errors, an adaptive neural network (NN) identifier is established by utilizing the measured input-output data of each joint module. The adaptive updating law guarantees that the NN weight error finally converged to a minimum neighborhood of zero. To ensure the optimality of system overall performance, the corresponding value functions reflecting the interconnectedness among each joint subsystem and manipulated object are constructed. Based on the idea of differential game, the coordination control problem of MRM system is transformed into a mixed nonzero-sum game problem among each joint module and the operated object. Next, by constructing a single critic NN with learning structure, the optimal value function is approximated to solve the event-based Hamiltonian equations, and then, the optimal control strategy of each player is obtained. Finally, the Lyapunov theory is used to analyze system stability, and the effectiveness of the presented method is reinforced by experimental results.

Game theory based vision impedance control for human-robot interaction.

Preference-based opponent shaping in differentiable games

Correction to: The Impacts of Reward Frequency and Reward Conditionality on Sustainable Use of Common Resources: a Stochastic Common Property Differential Game Model

This article investigates the distributed approximate optimal control problem for pursuit-evasion differential games (PEDGs) of multiagent systems (MASs). Initially, interactions between pursuer agents and the evader agents are formulated using a divide-and-conquer algebraic graph approach, where all agents desire to maintain cohesion with their teammates. Subsequently, a state event-triggered mechanism (ETM) is introduced to conserve communication resources. Meanwhile, a polymeric hierarchical sliding mode surface (HSMS) incorporating local neighbor errors is constructed such that the system response rate is improved. To enhance team coordination, a novel dynamic target allocation algorithm is designed to execute the rational allocation among pursuers. Furthermore, based on the adaptive dynamic programming (ADP) with a single-critic neural network (NN) architecture, the HSMS-based event-triggered optimal control policies are further designed via solving the coupling Hamilton-Jacobi-Bellman (HJB) equations. Finally, a simulation conducted in the representative two-pursuer-two-evader scenario is presented to validate the effectiveness of the proposed control scheme.

This article examines the coordination challenges within a supply chain comprising a manufacturer and a retailer, operating under the influence of environmental regulation instrument, namely green subsidy. A Stackelberg differential game model is developed to analyse equilibrium functions pertaining to environmental performance, retail pricing, and environmental perception levels in both integrated and decentralised channel settings. A two-part tariff contract is introduced for decentralised supply chain coordination, and a Nash bargaining model is subsequently employed to investigate the allocation of additional profits. Numerical simulations are conducted to assess the impact of the green subsidy coefficient and supply chain environmental standards on steady-state solutions and overall supply chain profit. The main findings underscore the flexible pricing strategies adopted by the supply chain in both integrated and decentralised settings. And the study indicates that the overall performance of the former surpasses that of latter. A two-part tariff contract emerges as a pivotal instrument for achieving effective supply chain coordination. Moreover, green subsidy coefficient exhibits a positive influence on enhancing coordination capability, whereas the supply chain environmental standard acts as a deactivator.

This paper presents a novel approach to address the problem of intercepting non-cooperative targets with multiple satellites in Earth orbit. The multi-satellite interception problem is formulated as a multi-player pursuit–evasion game that explicitly accounts for stochastic disturbances and control constraints. By combining differential game theory with stochastic optimization techniques, the paper derives optimal interception trajectories that ensure safety and performance under modeling uncertainties. A linear exponential quadratic cost functional is established, and corresponding Nash equilibrium strategies are obtained to determine the optimal control laws. Numerical simulations validate the effectiveness and robustness of the proposed approach in achieving reliable interception performance.

This paper investigates the non-zero-sum stochastic differential game problem of optimal asset-liability management (ALM) between insurers and reinsurers. Combining the investment- reinsurance problem with the ALM, this paper innovatively studies the investment-reinsurance ALM non-zero-sum game within the framework of the constant elasticity of variance (CEV) model under common shock dependence. We allow the insurance companies to purchase proportional reinsurance from reinsurance companies, and both companies can invest in a financial market composed of one risk-free asset and one risk asset whose price process follows the CEV model. By innovatively introducing a class of stochastic liability processes associated with the volatility of risky assets, we obtain the dynamic evolution of net wealth. Using stochastic control theory, we obtain an explicit expression of Nash equilibrium strategy for the problem and a closed-form expression for the corresponding equilibrium value function by maximizing the expected utility from the terminal net wealth. Finally, we illustrate the effect of relevant parameters on the optimal investment-reinsurance strategy by means of numerical examples.

The retrieval of end-of-life products is a critical component of closed-loop supply chain (CLSC) remanufacturing, yet achieving efficient recycling remains challenging due to coordination barriers between supply chain members. To address this issue, this study investigates the collaboration problem in end-of-life product collection within a CLSC consisting of a manufacturer and a retailer. The retailer is responsible for collecting end-of-life products, while the manufacturer may provide support through two alternative cooperation modes: fund cooperative and labor cooperative. Using the differential game approach, we develop equilibrium strategies under three scenarios—non-cooperation, fund-assistance cooperation, and labor-assistance cooperation. The analytical results show that cooperative collection strategies not only increase the recycling rate but also yield Pareto improvements, benefiting both the manufacturer and the retailer. Among the two cooperation modes, the labor cooperative achieves higher collection rates and greater joint profits than the fund cooperative. When considering heterogeneous collection costs between the manufacturer and retailer, the fund-assistance mode becomes more favorable for the manufacturer only when its collection cost substantially exceeds that of the retailer. Furthermore, we explore the combined implementation of fund and labor cooperative programs, revealing their potential to further enhance collection efficiency and overall profitability. This study contributes to the CLSC literature by introducing a dynamic differential game framework to model cooperative collection behaviors and provides actionable managerial implications for promoting manufacturer participation in used-product retrieval and fostering coordinated development across CLSC enterprises.

To meet the sustainable development goals of the aviation industry, promoting digitalized all-electric aircraft (AEA) is a critical path. However, during the dynamic popularization process of digitalized AEA, the interests among manufacturers, airlines, and governments vary, coupled with a notable time delay in digitalized technological R&D and market promotion. Therefore, this study establishes differential game models for popularizing AEA and investigates dynamic optimal strategies of potential benefits, levels of digitalized R&D, consumer preferences, and market demand, under three game modes: Nash non-cooperative, cost-sharing, and collaborative cooperation. The research finds that: (1) When the promotion cost of AEA is lower than a certain threshold, the cost-sharing model can effectively enhance digital R&D. (2) In the case of ignoring time lag, the initial value of the battery life level and consumer preference becomes the decisive factor that significantly affects its dynamic evolution trajectory. Under the cost-sharing model, the battery life level and consumer preference reached 107.13 and 15.26, respectively. This is significantly higher than the collaborative model and the NASH non-cooperative model. (3) When the delay effect exceeds the thresholds of 4.58 and 5.49, respectively, the Nash non-cooperative model becomes the most effective promotion model. This paper provides an important decision-making reference for promoting the digital transformation and sustainable development of the aviation industry.

Long-Time Behavior of Zero-Sum Linear-Quadratic Stochastic Differential Games

In this paper, we present a numerical solution for an open-loop Nash differential game modeling competition between two firms. Using the fourth-order Runge-Kutta method, we computed the numerical solution and analyzed the stability of the open-loop Nash equilibrium. Additionally, we examined the uniform convergence of the solution. Finally, an illustrative example is presented to clarify the results, accompanied by figures to illustrate the findings.

Considering a low-carbon operating environment featured by consumer low-carbon preference (CLP) and hybrid carbon policies, we construct a green brand supply chain consisting of a dominant manufacturer and two retailers. Under the green brand (GB) strategy, the manufacturer achieves green production through green technology and recycling, and the two retailers conduct green marketing activities. We adopt the differential game approach to characterise the dynamics of the green brand (GB) image, and fully consider the possibility of alliance between competing retailers and the consequent power structure shift. By analysing the four models, the results show that only when the retailer alliance has equal bargaining power with the manufacturer, i.e. the Nash power structure, competing retailers always have incentives to form an alliance. Under other power structures, the market condition for retailers to ally depends on the marginal effect of green production on GB image. For the manufacturer, the emergence of the retailer alliance may lead to positive outcomes. Under certain conditions, each player has the highest profit under the Nash power structure. In addition, we examine the impacts of the dynamic operating environment on equilibrium solutions and retailers’ alliance behaviour.

A Stochastic Differential Oligopoly Game for Housing Characteristics in Investment Decisions