Search Results (124)

To address three major industry pain points—the lack of quantified profit-sharing standards in the two-stage pricing under the separation model of military research and production, the absence of stable Nash equilibrium in single-layer synchronous optimization, and insufficient incentives for full-cycle process optimization in design units—this paper constructs a two-level Stackelberg leader–follower game model with the general contracting unit as the leader and design and general contracting units as followers. This aligns with current prototype incentives and phased pricing policies for production rewards and penalties. At the theoretical level, it improves the complete proof system for the two-stage concave profit two-level Stackelberg Nash equilibrium, distinguishes the mathematical differences in equilibrium existence between sequential decision-making and synchronous optimization, and extracts general rules for phased differentiated profit sharing: high-innovation segments should be allocated more profit weight; simply maximizing total alliance profit may cause imbalanced interests, while introducing a minimum net profit-weighted objective can achieve Pareto improvements without profit loss. This conclusion can be applied to multi-stage general contracting scenarios across industries, such as EPC and military–civil collaborative innovation, enriching the basic theory of profit sharing and hierarchical games. Theoretically, the existence of the lower-level Nash equilibrium is proven using Brouwer’s fixed-point theorem, and combining it with the strictly monotonically decreasing feature of the best response function, uniqueness of the equilibrium is derived. Multiple sets of differentiated initial values are simulated to rule out multi-equilibrium bifurcation risk. The model incorporates the military’s reward and penalty policies as rigid exogenous constraints, sets dual individual rationality constraints of ‘cooperative profit greater than baseline profit with no allocation, and both parties’ net profit non-negative,’ and introduces differentiated cost-reduction efficiency and quadratic increasing effort costs to characterize the heterogeneous input of the two types of development entities. For models with piecewise nonlinearity and multi-constraint nonconvex structures, this paper modifies the standard PSO into a Bi-PSO solving framework through hierarchical temporal adaptation. It does not innovate the underlying particle update mechanism and is only used to match the sequential decision order of the leader–follower game. By comparing five algorithms—IPM, GA, SA, DE, and adaptive PSO—through 20 repeated simulations: gradient-based interior point methods easily get stuck in locally invalid solutions that violate cooperation thresholds; differential evolution has the best numerical global search performance, but all general evolutionary algorithms optimize allocation and effort variables simultaneously, disrupting the Stackelberg hierarchical timing. Only Bi-PSO maintains consistent game logic. Using a pricing case for a certain type of equipment and jointly calibrating all parameters with policy documents, three simulation scenarios were set up: no allocation, equal 50/50 split, and single-layer profit maximization. Under the no-allocation mode, R&D investment from the design unit drops to zero and alliance benefits plummet; a blanket equal split ignores differences in technical contributions across two stages, leading to clear efficiency losses; single-layer optimization only pursues total profit maximization, causing a severe imbalance in profit distribution. The two-layer basic framework can achieve the upper limit of alliance benefits, and by adding a weighted optimization goal that considers both total profit and cooperation fairness, it can achieve equal net profits for both parties without reducing overall profit. Through single-parameter sweeps and two-factor heatmap simulations, the study further revealed the coupled effects of main party efficiency and mass production rewards and penalties on equilibrium input and optimal sharing ranges. A robust check was performed by replacing the logarithmic concave output function, producing a standardized allocation range resilient to parameter perturbations: optimal split for the prototype stage is 0.4–0.6 for the design unit, and for mass production stage 0.7–0.9. The findings suggest that high-contribution stages in multi-phase collaboration contracts should receive more benefits, and a weighted fairness objective can achieve Pareto improvements. These conclusions can extend to multi-stage collaboration scenarios such as EPC and military–civilian cooperation. Theoretically, this research further completes the equilibrium proof system for two-party concave payoff two-layer games, providing a new reference for the theory of phased differentiated benefit-sharing contracts in the military sector. Methodologically, it proposes a two-layer intelligent solving tool adapted to leader–follower sequential decisions, effectively mitigating issues where single-layer model equilibria fail or analytical algorithms struggle with multi-constraint nonconvex games. The results can provide quantitative support for the military, general contracting unit, and design unit in drafting equipment incentive pricing contracts and managing full-cycle cost collaboration. Full article

Against the backdrop of the “Rural Revitalization” strategy, contract farming has emerged as a new mechanism and a growing research focus. This study employs differential game theory to systematically analyze the dynamic decision-making and operational performance differences among supply chain members by comparing scenarios with and without research institution participation, as well as different government subsidy policies. The innovative contribution lies in introducing a dual-effect index to capture consumers’ quality utility from agricultural products and psychological utility derived from supporting farmers. Furthermore, we explore the synergistic effects of research institutions’ technological empowerment and government subsidy strategies on supply chain efficiency. Through the above research and analysis, the study draws the following conclusions: (1) The collaborative agricultural supporting model significantly enhances both farmer yields and overall supply chain profits through technology diffusion and brand premium effects. However, revenue distribution conflicts may lead enterprises to limit their cooperation depth. (2) Government subsidies can alleviate benefit allocation conflicts. Subsidizing a research institution proves more effective at stimulating long-term technological dividends, while subsidizing an enterprise primarily ensures short-term market stability. (3) Brand advantage, technological leadership, and consumer preferences exhibit nonlinear driving effects on supply chain efficiency. Specifically, the positive impact progressively strengthens with wider brand advantages, greater technological leadership, and more intense consumer preferences. The study provides a theoretical foundation for tripartite collaboration, revealing the pivotal role of policy precision and dynamic benefit allocation equilibrium in agricultural supply chain upgrading. Full article

The Target–Attacker–Defender (TAD) pursuit–evasion game is a core challenge in multi-agent cooperative control, yet real-world settings involving dynamic team scaling and strict energy constraints remain largely unaddressed. When scalable shared-parameter policies are adopted to cope with the varying number of agents, severe policy homogeneity emerges, preventing effective division of labor. This paper proposes a Hierarchical Heterogeneous Multi-Agent Proximal Policy Optimization (HH-MAPPO) framework to resolve these challenges. Both levels employ actor–critic networks with Role-Aware Embedding (RAE). In this mechanism, each agent is assigned a unique, learnable role embedding derived from its identity. These embeddings serve as conditioning inputs to the shared policy network, enabling it to generate differentiated behaviors and effectively mitigating policy homogeneity. The upper-level policy determines the number of defenders to deploy and assigns interception targets, while the lower-level policy handles continuous control of each defender and the ground moving target (GMT). This hierarchy resolves dynamic observation spaces via a target-matching mechanism, where each defender’s observation includes only its own state and its assigned attacker’s state, keeping observation dimension constant. Experiments in a 3D TAD simulation with continuous attacker arrivals and energy-constrained defenders show the following: (1) HH-MAPPO achieves superior interception performance compared to baseline methods in both symmetric and asymmetric scenarios; (2) ablation studies confirm RAE increases policy diversity, raising Sequence-Based Action Dissimilarity (SBAD) by 15.5%; and (3) Pareto analysis demonstrates a superior performance–energy trade-off, maintaining about 70% interception rate even under an extreme energy cap (E = 30). Full article

The spacecraft reachable domain has become increasingly important for orbital game analysis due to growing on-orbit activities such as servicing, debris removal, and space situational awareness. This paper provides a comprehensive review of reachable domain theory and its applications in orbital games. A unified mathematical framework is established through three complementary classification dimensions: spatial attributes that distinguish absolute from relative reachable domains, temporal attributes that differentiate free-time from fixed-time reachable domains, and informational attributes that contrast deterministic and predictive reachable domains. Solution methods are systematically reviewed according to this taxonomy, covering analytical and semi-analytical methods, numerical optimization approaches, and geometric and sampling methods for spatial-scale reachable domains, as well as linearized ellipsoidal approximation, exact envelope determination, and fast analytical approximation for time-scale reachable domains. Applications are examined through three representative scenarios: one-on-one pursuit-evasion games, multi-agent cooperative games, and threat-avoidance and defense games. Key limitations of existing approaches are identified, including modeling fidelity, computational efficiency, and scalability under uncertainty. Future research directions are outlined to address these challenges. Full article

With the increasing penetration of distributed renewable energy, coordinated operation between distribution networks and multiple integrated energy microgrids (IEMs) has become increasingly important for improving operational economy and maintaining system security. To address the insufficient integration of network constraints, P2P energy sharing, and contribution-based benefit allocation, this paper proposes a hybrid game-based optimal scheduling model for multi-IEM systems under distribution network constraints. In the upper level, a Stackelberg game is established between the distribution system operator (DSO) and the multi-IEM alliance to coordinate electricity trading and distribution network operation. In the lower level, an asymmetric Nash bargaining-based cooperative game is developed to enable peer-to-peer (P2P) energy sharing and allocate cooperative benefits according to the actual energy-sharing contributions of individual IEMs. Furthermore, a distributed solution framework combining the Success-History Adaptive Differential Evolution (SHADE) algorithm and an improved alternating direction method of multipliers (ADMM) is adopted to preserve data privacy and improve computational efficiency. Case studies based on the modified IEEE 33-bus distribution system demonstrate that the proposed method eliminates voltage violations and reduces network losses by 9.0%. Meanwhile, the proposed cooperative mechanism decreases the total operating cost of the IEM alliance by 7815.8 CNY and yields a more contribution-consistent profit allocation among participating microgrids. In addition, the improved ADMM reduces computation time by 42.1% compared with the conventional serial ADMM. The results demonstrate the effectiveness of the proposed method in enhancing distribution network security, promoting renewable energy sharing, and improving the economic performance of multi-IEM systems. Full article

The rapid growth of the sharing economy has improved resource utilization in car-sharing, yet it has also sharpened market competition and diversified user demand. A persistent obstacle is the low coordination efficiency between asset-heavy operating companies and traffic-driven platforms, whose misaligned objectives waste social resources. This paper uses differential game theory to analyze their dynamic coordination strategies and benefit allocation mechanisms. The Nerlove–Arrow model captures the evolution of brand goodwill, while the company’s decisions on station layout, vehicle dispatch, and pricing, together with the platform’s advertising investment, form the core decision variables in a two-party game framework linking the asset side and the traffic side. Compared with the non-cooperative Nash equilibrium, the cooperative mode removes the double marginalization effect, strengthens the investment incentives of both parties, and raises the system’s steady-state goodwill and total profit, achieving a Pareto improvement. To ground the cooperative framework in rigorous theory, we supply a verification theorem confirming that the linear candidate value functions satisfy the Hamilton–Jacobi–Bellman equations over the entire admissible state space. A formal proof of instantaneous rationality ensures that neither party falls into a cooperation trap on the horizon $[0,T]$, and the asymptotic stability of the steady-state goodwill trajectory is established. We further endogenize the revenue-sharing coefficient through a generalized Nash bargaining model that admits asymmetric bargaining structures, and introduce a Stackelberg leadership benchmark as a third comparative regime. Sensitivity analyses with respect to the discount rate and user heterogeneity confirm the robustness of the findings. A dedicated discussion section bridges the gap between idealized parameterization and data-driven calibration, describing practical pathways via A/B testing, user churn metrics, and econometric estimation of demand parameters. The results offer a scientific decision-making reference for strategic cooperation in the car-sharing industry. Full article

We present a review on the application of the mathematical models for research on social processes, social structures, and actors in social systems. The scope of the review is not restricted to the classical applications of mathematics such as theory of probability, statistics, stochastic processes, differential equations, and game theory. We also discuss applications of the theory of networks for social network analysis and the numerical research on dynamics of social systems. The number of these applications has increased very fast in recent years. Special attention is given to the results from the area of sociophysics, where mathematical methodology is used to analyze social systems in cooperation with the models and concepts of physics. Another special topic in his review is connected to the results from econophysics, where the mathematical methodology and theories and methods of physics are used in the studies on the dynamics of economic systems. In addition, we give several examples for the application of mathematical methods to social systems: (a) application of difference equations to model the flow of substances in channels of networks; (b) analytical solution of nonlinear equations connected to the model of waves of popularity; (c) numerical results of the waves of popularity in a model that accounts for the change in the opinion of the supporters of the ideas for positive or negative popularity of a person, material item, or a piece of information (idea, theory, ideology, etc.) In the last case, we illustrate the effectiveness of the numerical analysis to discover new effects on the studied social system. The review ends with a large list of references. These references can be used as a guide of the way of new researchers to the large field of mathematical social dynamics. Full article

We study a continuous-time cooperative differential game of pollution control in which the pollution stock accumulates emissions and affects long-run welfare. The key feature is a one-time random increase in the public damage weight, interpreted as a regime shift in environmental policy, social damage assessment, or regulatory pressure. Using dynamic programming, we characterize the grand-coalition feedback solution from the Hamilton–Jacobi–Bellman equations and derive closed-form expressions for cooperative emissions, pollution dynamics, regime-specific steady states, and transition paths. Under emission caps, we construct the coalition characteristic function using a conservative worst-case benchmark for outsider behavior rather than an unlimited-pollution assumption. For payoff allocation, we derive a dynamic payment schedule that implements the Shapley allocation along the stochastic pollution path and keeps the remaining payoff consistent with the corresponding continuation game. Finally, we extend the framework to a threshold-triggered shifted-exponential switching mechanism. This extension gives a computable objective for the optimal threshold-hitting time and clarifies how the pollution threshold and switching hazard can be interpreted as policy-relevant indicators of regulatory or ecological regime change. Full article

The realization of the value of health big data relies on the coordinated cooperation among patients, the government, and data users. Enhancing the symmetry and balance between patient participation and the compliant use of data by data users is a critical link. This paper constructs a tripartite evolutionary game model and employs MATLAB R2023a simulation to analyze the impact of factors such as initial willingness, compliance costs, and penalties for violations on the strategic choices of the game players and the evolution of the system. The findings reveal that: (1) Patient participation is a key condition for achieving an ideal equilibrium in the system. (2) The data service income from participating in data provision and the costs associated with privacy breaches are critical factors influencing patients’ strategic choices. (3) Penalties for violations are a crucial factor in ensuring that data users choose compliant utilization; however, when compliance costs are high, their constraining effect may be somewhat diminished. (4) Enhancing regulatory efficiency is the future direction for government departments. Based on these findings, countermeasures and suggestions are proposed, including trust building, technological innovation and differentiated supervision, and constructing trusted data spaces, to provide references for health big data governance. Full article

High-penetration renewable energy significantly increases uncertainty, dynamic network coupling, and the need for secure and coordinated smart-grid dispatch. To address the limitations of conventional forecasting-based and static graph-based methods, this paper proposes a unified dispatch framework that integrates topology-informed dynamic graph learning, privacy-aware multi-agent symbiotic reinforcement learning, and structural causal intervention analysis. The dispatch problem is formulated as a constrained partially observable stochastic game, in which multiple agents coordinate generation adjustment, reserve allocation, and congestion-aware corrective actions under engineering constraints. A physics-informed dynamic graph convolutional module captures both fixed physical topology and stress-dependent operational couplings, while a KL-regularized multi-agent reinforcement learning scheme improves cooperative task allocation under renewable fluctuations. Federated optimization with Rényi differential privacy is introduced to protect sensitive local operational information during training. In addition, a structural causal module provides intervention-based interpretation of how wind variation, load escalation, and line stress affect dispatch cost, congestion risk, and renewable curtailment. Experiments on a public-trace-driven benchmark based on a modified IEEE 30-bus system show that the proposed method achieves the best overall performance among the compared baselines, reducing dispatch-cost RMSE to 3.82, locational-price MAE to 2.95, renewable curtailment to 4.8%, and the constraint-violation rate to 0.30%. Overall, the framework shows favorable performance on the test benchmark, provides post hoc intervention-based interpretation of dispatch outcomes, and is evaluated under a reproducible benchmark construction and assessment protocol. Full article

For the complex problems of multi-aircraft cooperative game guidance with No-Fly Zone (NFZ) avoidance and cross-task constraint propagation, a deep deterministic policy gradient algorithm with temporal awareness and priority cooperative optimization (TP-MADDPG) is proposed. Based on the three-body cooperative guidance, a new coupled guidance task is formed by adding the NFZ avoidance constraint. At the same time, considering the constraint compatibility problem in dynamic task switching, the cooperative aircraft are modeled as independent agents with differentiated policy networks. First, a nonlinear kinematic model of the three-body game constructed by Evader–Pursuer–Defender is established. And four complex constraint conditions, namely homing guidance, NFZ avoidance, collision avoidance, and cooperative guidance, are modeled separately. Secondly, the Long Short-Term Memory-based (LSTM) Actor–Critic framework is proposed to dynamically capture the evolution patterns of adversarial scenarios by mining hidden correlations in historical state-action sequences. This enables smooth policy transitions between the cooperative guidance phase and subsequent homing guidance phase, effectively addressing the challenges of environmental non-stationarity and temporal task dependencies. Then, a priority-driven adaptive sampling mechanism is proposed along with a heterogeneous roles cooperative reward function to specifically address credit assignment imbalance and sparse reward problems, respectively. The sampling mechanism capitalizes on the efficient retrieval properties of SumTree data structures while integrating bias correction techniques to expedite policy gradient convergence. The reward function utilizes the reward shaping method to formulate cooperative reward components that explicitly capture behavioral correlations among agents. Finally, simulations show that the proposed method significantly outperforms multi-agent reinforcement learning baselines, effectively improving the performance of cooperative game guidance under complex constraints. Full article

E-commerce platforms are increasingly adopting cross-platform targeted advertising strategies, and the design of such strategies warrants attention. Focusing on cooperation between e-commerce and social media platforms, this study considers targeting precision, advertising intensity, privacy concerns and social utility on the effectiveness of targeted advertising. Using a game-theoretic model, we examine the decision between single- and cross-platform for e-commerce platforms in fully and partially overlapping user groups. The main findings indicate that (1) the social utility of social media platforms is a key factor in implementing cross-platform targeted advertising; (2) cross-platform targeted advertising is not always the optimal choice for e-commerce platforms; and (3) low-precision cross-platform strategy achieves three-party optimum in fully and partially overlapping user groups. The implications of the main findings include: (1) e-commerce platforms should prudently use social media platforms instead of relying excessively on their traffic; (2) e-commerce platforms should not regard cross-platform cooperation as the default option but as a differentiated, situation-specific decision; and (3) e-commerce platforms should promote co-creation of value and proprietary data accumulation when cooperating with social media platforms. The findings can help e-commerce platforms to choose proper targeted advertising strategy in practice. This study also provides a theoretical supplement for cross-platform targeted advertising research. Full article

In high-density urban environments, outdoor kindergarten spaces are vital for children’s cognitive and social development, yet their design within constrained urban greenery poses a significant challenge. This study investigated how these environments support development through autonomous play. Conducted as a case study in three Beijing kindergartens, it employed a framework analyzing ten environmental elements across four dimensions: terrain space (e.g., open space, slopes), game facilities (fixed and movable), loose materials, and natural elements (water, plants). Behavioral observations were used to examine associations between these elements and children’s play behaviors. The findings suggest that diverse, naturalized, and adaptable combinations of elements may best foster autonomous play. While functional play was predominant, our analysis identified that a core combination of rigid fixtures, shielded places, and loose materials appears to optimally support this play type, which is primarily linked to solitary play. By strategically supplementing this core with elements like moving fixtures and loose objects, the environment can further encourage constructive, dramatic, and exploratory play—forms that show stronger associations with cooperative group play. This reveals a potential pathway through which sequenced environmental provisioning might scaffold the progression from individual to social play, thereby fostering socio-cognitive growth. Consequently, the study proposes three exploratory design principles: the differentiated allocation of elements to target specific play behaviors, deliberate naturalization of the setting, and incorporating dynamic adjustability for flexibility. These hypothesis-generating strategies aim to inform the design of kindergarten outdoor spaces, offering practical guidance for creating more sustainable and child-inclusive urban communities, though their generalizability requires further cross-context validation. Full article

This study examines the optimal design of low-carbon policies for governments, firms, and consumers within a unified analytical framework. We develop a three-stage game-theoretic duopoly model with differentiated products, green R&D, and knowledge spillovers to analyze the effects and implications of low-carbon policies in a polluting industry. The analysis encompasses both Cournot and Bertrand competition under commitment and non-commitment regimes, as well as non-cooperative and cooperative R&D structures. Specifically, we (i) quantify the impacts of low-carbon policies on R&D, emissions, profits, and welfare across alternative competition modes, policy-timing regimes, and R&D organizations; (ii) examine the roles of key policy parameters across all scenarios; and (iii) provide an integrated and intuitive interpretation of the underlying economic mechanisms. Full article

The promotion of shore power is a key pathway for reducing port-related emissions and achieving sustainable maritime development. This study analyzes the strategic interactions among governments, ports, and shipping companies by constructing a tripartite evolutionary game model. Specifically, it addresses three core questions: (1) how stakeholders’ initial intentions and strategic choices influence the system’s evolutionary path and eventual equilibrium; (2) how critical parameters—including subsidies for shore power infrastructure, wind turbine installation, and ship retrofitting, as well as electricity price support, carbon pricing, and policy implementation costs—shape the dynamics of the system and the equilibrium strategies of the three parties; and (3) how heterogeneity in national energy mixes, particularly the roles of wind turbine, affects decision-making behaviors across different countries. Simulation experiments are conducted to explore the effects of varying policy interventions and energy conditions on the stability of cooperative strategies. The results provide insights into the design of differentiated policy instruments that promote shore power adoption while accounting for the structural characteristics of national energy systems. This research enriches the theoretical application of evolutionary game theory to maritime sustainability and offers practical guidance for governments and stakeholders in advancing decarbonization in the port and shipping sectors. Full article