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Based on the carbon emission policy and low-carbon capacity sharing, this paper studies the optimal product pricing and capacity matching strategies of competition and cooperation between two companies in a duopoly market. And the optimal strategies with or without capacity sharing are discussed. In addition, a differential game model based on carbon emission constraints, low-carbon capacity sharing, and delayed decision-making is established. We also study the evolution of duopoly enterprises from the initial state to the equilibrium state. Besides, the influence of delayed decision variables, price adjustment speed and decision weights on the stability of the game system is analyzed. The results show that the government can effectively intervene and guide the enterprises' low-carbon production strategy through carbon emission restriction policy. Low-carbon capacity sharing is beneficial to both enterprises, and the demand side benefits more than the supply side. We discuss the impact of stable and unstable systems on enterprises' decisions. When the system is stable, enterprises can converge from different initial game states to equilibrium states. The adoption of a delay strategy will not affect the tendency of convergence to the equilibrium. Furthermore, the system using the variable feedback control method is of higher stability. [ABSTRACT FROM AUTHOR]

We consider a low-carbon tourism supply chain (LTSC) network composed of two low-carbon scenic spots (LSSs) and two travel agencies (TAs). The LSSs as leaders and the TAs as followers play Stackelberg games in the vertical direction. Meanwhile, the competing LSSs play Nash game to determine emission reduction efforts and ticket prices, and the competing TAs play Nash game to conduct retail prices. We establish four kinds of differential game models with different LTSC network structures, derive the equilibrium decision-makings of the LSSs and TAs, and explore the influences of competition or cooperation at vertical and horizontal level of the LTSC network members on the equilibrium solutions of the LSSs and TAs, low-carbon level of tourism products and performance of the LTSC network. Then, we find that the low-carbon preference of tourists improves the low-carbon level of tourism products, while the competition intensity between the LSSs leads to the decrease of the low-carbon level of tourism products. In addition, the results indicate that horizontal or vertical cooperations among the LTSC network members are not always conducive to the performance of the whole LTSC network. Only when the sensitivity of tourist demand to retail price meets certain conditions, these cooperations can develop tourism products with higher low-carbon level and bring higher revenues. [ABSTRACT FROM AUTHOR]

Uncertainty in the evolution of opponent behavior creates a non-stationary environment for the agent, reducing the reliability of value estimation and strategy selection while compromising security during the exploration process. Previous studies have developed various uncertainty quantification techniques and designed uncertainty-aware exploration methods for multi-agent reinforcement learning (MARL). However, existing methods have gaps in theoretical research and experimental verification of decoupling uncertainty between opponents and environment, which can decrease learning efficiency and lead to an unstable training process. Due to inaccurate opponent modeling, the agent is vulnerable to harm from opponents, which is undesirable in real-world tasks. To address these issues, this study proposes a novel uncertainty-guided safe exploration strategy for MARL that decouples the two types of uncertainty originating from the environment and opponents. Specifically, we introduce an uncertainty decoupling quantification technique based on a novel variance decomposition method for action-value functions. Furthermore, we present an uncertainty-aware policy optimization mechanism to facilitate safe exploration in MARL. Finally, we propose a new adaptive parameter scaling method to ensure efficient exploration by the agents. Theoretical analysis establishes the proposed approach's convergence rate, and its effectiveness is demonstrated empirically. Extensive experiments on benchmark tasks spanning differential games, multi-agent particle environments, and RoboSumo validate the proposed uncertainty-guided method's significant advantages in attaining higher scores and facilitating safe agent exploration. [ABSTRACT FROM AUTHOR]

Keeping an intelligent agent under surveillance with an autonomous robot as it travels in the environment is one of the essential tasks in robotics. This work addresses a pursuit-evasion surveillance problem between two Differential Drive Robots moving on a plane without obstacles. One of them, the pursuer, is equipped with a limited field-of-view sensor and aims to maintain the other robot (evader) inside its detection region for as much time as possible. On the contrary, the evader wants to escape as soon as possible. Unlike prior studies that address the same problem, based on heuristic approaches, we frame it as a zero-sum differential game, and we use differential game theory to find a solution with theoretical guarantees. In particular, we compute the players' time-optimal motion strategies to achieve their goals near the end of the game and provide analytical expressions describing them. These strategies are in Nash equilibrium, meaning that any unilateral deviation by a player from these strategies does not provide such player benefit toward its goal. Our analysis exhibits the existence of a Transition Surface in which the evader switches its controls. We also determine the game's winner based on the players' initial configurations and velocities. [ABSTRACT FROM AUTHOR]

It develops a differential game model to investigate the issue of fuel consumption reduction investments and marketing cost coordination in a fuel vehicle supply chain to reduce carbon emissions. In this case, consumers have low fuel consumption preferences, and the government subsidizes ex-ante or ex-post the investments of supply chain firms. Accordingly, we propose two dynamic equations for fuel consumption reduction and goodwill of vehicles, as well as a demand function that accommodates consumers' low fuel preferences and automotive goodwill. Then, we analyze the different government subsidy models and supply chain marketing cost-sharing in four cases. The results show that both ex-ante and ex-post subsidies provide adequate incentives for fuel consumption reduction investments, but which subsidy policy is preferable depends on the parameter values; cost-sharing leads to increased marketing effort by retailers rather than reduced it; there is no correlation between the fuel consumption reduction investments and marketing effort; and the sharing contract of marketing costs leads to Pareto improvements that include the supply chain and society. [ABSTRACT FROM AUTHOR]

With the rapid growth of social media and live-streaming technology, live-stream selling has become integral to the digital economy. Using differential game theory, this paper examines how fairness concerns impact the profits of internet celebrities and brand manufacturers under the "pure commission" model. We analyzed no fairness concern, gap fairness concern, and self-due fairness concern models, to investigate the optimal decisions and corresponding profits for an internet celebrity and a brand manufacturer. The results show that the internet celebrity earned the highest profits with low commission rates under the self-due fairness concern model, whereas higher commission rates yielded higher profits for the internet celebrity under the gap fairness concern model. Simultaneously, fairness concerns significantly affected the cooperation stability and long-term benefits, motivating the internet celebrity to maintain efficient collaborations with the brand manufacturer. Furthermore, the self-due fairness concern model was more practical than the gap fairness concern model. [ABSTRACT FROM AUTHOR]

This paper explores the optimal management of defined contribution pension plans under multiple risks and strategic interactions using a stochastic differential game between two managers, accounting for model misspecification. Managers who are both competitive and averse to ambiguity can invest in a financial market that contains a risk-free asset, an inflation-linked bond, a market index, and a stock. These managers may have varying levels of aversion to ambiguous risks. The inflation index is computed using the Fisher equation. The market index and stock prices are described by a class of non-Markovian multivariate stochastic covariance models, with the market risk prices being reliant on a multivariate affine-diffusion factor process. Each manager aims to maximize the expected utility of inflation-adjusted terminal wealth relative to their competitors under the worst-case scenario of different measures, ensuring that the investment strategy is robust to model uncertainty. The robust Nash equilibrium investment strategies' explicit expressions, density generator processes under the well-defined worst-case scenarios, and corresponding value functions are derived using a backward stochastic differential equation approach to address this robust non-Markovian game. The admissibility of the robust equilibrium policies is confirmed under certain technical conditions. Finally, we provide numerical examples to demonstrate the effects of model parameters on robust investment policies and clarify the economic significance of our theoretical findings. [ABSTRACT FROM AUTHOR]

The sequence leading to conflict, as specifically in July 1914, reveals the changes in each protagonist's propensity to wage war. To characterize its timing, escalation is viewed as a viability (differential) game in progress under the protagonists' actions. The chronicle of confrontations and accommodations allows for estimating a responsiveness structure at the heart of the game. As the moves, which are the controls, are observed, and not the state variables, the game reduces to the minimization of a viability criterion under dynamics. Early estimation of the responsiveness structure operationalizes the procedure for crisis management and provides a way to anticipate the consequences of possible moves in escalations to conflict. The application to July 1914 shows the strength of the method. [ABSTRACT FROM AUTHOR]

China has emerged as a major global player in the pharmaceutical industry and a significant consumer of drugs. However, the rapid growth of the raw material pharmaceutical manufacturing sector has resulted in a substantial amount of waste generation, leading to environmental pollution issues. In order to mitigate the pollution problems associated with the pharmaceutical manufacturing industry, this research proposes a tripartite game model with two dynamic parameters and develops optimization models for green supply chains under three decision-making scenarios, aiming to contribute to the efforts for achieving low-carbon development. Additionally, the study considers the government's green subsidy system and incorporates the optimization of green pharmaceutical practices and goodwill of the enterprises as dynamic parameters. By employing differential game methods, equilibrium solutions among the three parties are determined. The research findings reveal that: firstly, the green pharmaceutical levels are equivalent among the three models; secondly, in scenarios where retailers receive subsidies and make decentralized decisions, API manufacturers achieve the highest level of goodwill and the supply chain profit is maximized; thirdly, when manufacturers have sufficiently high marginal profits, adopting a decentralized decision-making approach encourages retailers and enhances the overall goodwill level of the supply chain. This study provides valuable insights for the formulation of effective strategies and policies to promote sustainable development within the pharmaceutical industry. [ABSTRACT FROM AUTHOR]

In the paper, the formation control problem in unknown environments for networked multi-unmanned aerial vehicle systems (UAVSs) is resolved under a nonlinear differential game (NL-DDG) framework. The main challenge of this framework is how to obtain feedback Nash strategies, which typically overly rely on global information and cannot ensure the existence of Nash strategies in unknown environments. Toward this goal, we initially design collision avoidance rules to ensure the safety of each UAV. Subsequently, we utilize an inverse optimal control method to construct the NL-DDG that incorporates both formation control and collision avoidance costs, enabling the derivation of analytical forms of Nash strategies relying solely on local information and minimizing performance metrics. In addition, the existence of feedback Nash strategy can be guaranteed with an undirected and connected information topology, which represents the optimality for the UAVSs. Moreover, we analyze the stability of the closed-loop system. Finally, the simulation results validate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents { γ 1 , γ 2 , ... } under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid load frequency control (LFC) scheme is proposed for multi-area interconnected power systems to decouple the intricate double control objectives, by dividing all subareas into the responsible areas and the free areas. The LFC in the responsible area has the function of regulating both the local frequency and the tie-line power, while the control objective of the LFC in the free area is thus simplified to regulate the local frequency only. Then, addressing the complex network coupling and uncertain dynamics, an integrated LFC controller is proposed for the free areas, which consists of two parts, namely, the coupling attenuation baseline controller and the disturbance compensation controller. The coupling attenuation baseline controller satisfying the predefined bounded L2-Gain condition is derived based on the solution to a multi-player zero-sum differential game. Additionally, a novel generalized integral observer is designed to estimate the system's integrated disturbance, and the corresponding disturbance compensation controller is derived. After that, the ultimately uniformly bounded (UUB) stability of the integrated LFC controller combining baseline controller and disturbance compensation controller is proven rigorously. Finally, the performance superiority of the proposed hybrid LFC scheme is validated by the simulations in challenging operating modes. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to identify an optimal coefficient for evaluating a player's batting average, strike rate, and bowling average, aimed at achieving an optimal team score through dynamic modeling using a path integral method. Additionally, it introduces a new model for run dynamics, represented as a stochastic differential equation, which factors in the average weather conditions at the cricket ground, the specific weather conditions on the match day (including abrupt changes that may halt the game), total attendance, and home field advantage. An analysis of real data is been performed to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates a parameter-free H∞ differential game approach for nonlinear active vehicle suspensions. The study accounts for the geometric nonlinearity of the half-car active suspension and the cubic nonlinearity of the damping elements. The nonlinear H∞ control problem is reformulated as a zero-sum game between two players, leading to the establishment of the Hamilton–Jacobi–Isaacs (HJI) equation with a Nash equilibrium solution. To minimize reliance on model parameters during the solution process, an actor–critic framework employing neural networks is utilized to approximate the control policy and value function. An off-policy reinforcement learning method is implemented to iteratively solve the HJI equation. In this approach, the disturbance policy is derived directly from the value function, requiring only a limited amount of driving data to approximate the HJI equation's solution. The primary innovation of this method lies in its capacity to effectively address system nonlinearities without the need for model parameters, making it particularly advantageous for practical engineering applications. Numerical simulations confirm the method's effectiveness and applicable range. The off-policy reinforcement learning approach ensures the safety of the design process. For low-frequency road disturbances, the designed H∞ control policy enhances both ride comfort and stability. [ABSTRACT FROM AUTHOR]

We consider first-price auctions with independent and private valuations that have asymmetric valuation distributions and supports. We first show the existence of equilibrium in these auctions through a perturbation approach, thereby establishing that the limit of Bayesian Nash equilibria (BNE) of such perturbed auctions is indeed the Bayesian Nash equilibrium (BNE) of the limit auction with asymmetric supports. We then characterize this BNE and show that the ε -equilibrium (ε -BNE) of the auction with asymmetric supports is a BNE of "close" auctions with common supports. We then demonstrate some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we advocate for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin's canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimension n − 1 renders the optimal control problem easier to solve. The global solution of the optimal control problem is obtained and the synthesized optimal control law is in state feedback form. In this respect, two target sets are considered: a smooth circular target and a square target with corners. Closed-loop state-feedback control laws are synthesized that drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs' method for the solution of differential games, which entails dynamic programming (DP), working backward from the usable part of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions for optimality provided by Pontryagin's Maximum Principle (PMP). In this paper, the case is made for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems by way of the juxtaposition of the PMP and DP methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization70(10), 2131–2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results. [ABSTRACT FROM AUTHOR]

This paper studies a partially observed stochastic Stackelberg game with two followers, where state satisfies a linear stochastic differential equation of mean-field type, and cost functionals are quadratic. Using decomposition technique and backward separation approach, we derive the followers' optimal strategy. The leader focuses on an optimal control problem driven by a fully coupled mean-field forward–backward stochastic differential equation with conditional expectation. By layer-by-layer decomposition technique, some inequalities and Riccati equations, we not only get the existence and uniqueness of solution to the corresponding Hamiltonian system but also give a feedback form of Stackelberg solution. Finally, we tackle a government debt problem by the above theoretical results. [ABSTRACT FROM AUTHOR]

This paper primarily investigates a neural-network-based safe control scheme for solving the optimal control problem of continuous-time (CT) nonlinear systems with asymmetric input constraints under non-zero-sum (NZS) differential game scenarios. Initially, by constructing a novel non-quadratic function, the issue of asymmetric input constraints in the non-zero-sum differential game controllers is addressed. Subsequently, the safe Hamilton-Jacobi-Bellman (HJB) equation is derived from the direct integration of the control barrier function (CBF) into the traditional cost function, ensuring that the system states remain within a safe region. Then, the safe learning control scheme based on single critic neural network (NN) and adaptive dynamic programming (ADP) is proposed to approximate the optimal control strategy, differing from the dual-network update method commonly used in traditional ADP. Based on the constructed neural network weight adjustment rules, the optimal solution to the HJB equation can be derived within the safe learning control framework. Following this, Lyapunov's stability theory demonstrates that the errors in neural network weights and all signals within the closed-loop system are uniformly ultimately bounded (UUB). Finally, the effectiveness of the developed neural-network-based safe learning control method is validated through two simulation results. [ABSTRACT FROM AUTHOR]

This paper investigates a multi‐pursuer single‐evader nonzero‐sum differential game. Unlike traditional treatments, cooperative tracking is considered to design the optimal pursuit strategy by introducing a novel weighted topology and a scalar coupling gain. First, the distributed control strategy is designed based on the Hamilton–Jacobi–Isaacs (HJI) under the nonzero‐sum differential game. Then the modified LQR‐based cooperative tracking strategy is applied to achieve cooperativity among all the pursuers. The interception condition is also analyzed using a coupling gain based on the Laplacian matrix. Several illustrative examples under different scenarios are presented, and the results show that the pursuers with the proposed strategy can successfully intercept the evader. [ABSTRACT FROM AUTHOR]

The carbon quota allocation method serves as the foundation for the design of the carbon trading mechanism, which has a significant impact on supply chain production decisions and the operational efficiency of the carbon trading market. To analyze the behavioral decision problem of supply chain members under different carbon quota allocation methods, the low-carbon reference effect is introduced to characterize the effect of consumers' low-carbon preference on market demand. On this basis, three differential game models are constructed, namely, no emissions penalty, trading under the grandfathering principle, and trading under the benchmarking principle. The results indicate that the implementation of carbon trading policies enhances consumers' low-carbon reference levels, the carbon emission reduction levels of manufacturers, and the low-carbon publicity levels of retailers. Moreover, the enhancement of the low-carbon reference effect becomes a positive driver of profit growth. Manufacturers are observed to make more efforts in carbon reduction under the benchmarking principle compared to the grandfathering principle. In contrast, the level of low-carbon publicity by retailers remains unchanged. The above findings can provide a scientific basis for the decision-making of emission reduction in low-carbon supply chain enterprises, which has certain theoretical significance. [ABSTRACT FROM AUTHOR]

To deal with the offense-defense confrontation problem of multi-missile cooperative intercepting a high-speed and large-maneuvering target, a differential game-based cooperative interception guidance law with collision avoidance is proposed, in which the offense-defense parties are the incoming target and the interceptors, respectively. Given that both offense-defense parties have uniformly decreasing speeds and first-order biproper dynamics, the relative motion models among the offense-defense parties are established, and the performance indices of the target and the interceptors are proposed. After that, the cooperative interception guidance law with collision avoidance is derived based on a differential game. The guidance law considers the effects of speed variations and rudder layouts on the motions of both offense-defense parties, ensuring excellent algorithmic real-time property and interception accuracy while introducing inter-missile collision avoidance constraints. In addition, the parameters of the target performance index are set according to the target acceleration information estimated by the interceptors. The simulation results verify the effectiveness of the guidance law designed in this paper, under various three-to-one scenarios, the interceptors could achieve collision-free interceptions with the interception accuracy of less than 5 m and the interception time difference of less than 0.1 s. [ABSTRACT FROM AUTHOR]

The paper focuses on a three-echelon supply chain consisting of a green supplier, a green manufacturer and a retailer. Considering the dynamic changes in product greenness, differential games are employed to investigate product prices, green efforts and coordination in the system. By comparing equilibrium results in various scenarios, it is found that supply chain decision-makers invest more in green efforts to maximize their profits under centralized decision, while they select pricing tools under decentralized decision. Importantly, when a local alliance composed of the supplier and manufacturer supplies products to the retailer at a flexible and dynamic wholesale price, the two-part tariff contract can fully coordinate the three-echelon green supply chain. The Rubinstein bargaining model is then used to optimize the profit allocation within the alliance. Additionally, the impact of green effort efficiency on decision-making is explored, revealing that green efforts depend on the efficiency of green efforts, while product pricing has nothing to do with the efficiency of efforts. The paper also points out that decentralization increases the probability of the supplier engaging in more green efforts than the manufacturer. [ABSTRACT FROM AUTHOR]

This paper introduces two new identification methods for linear quadratic (LQ) ordinal potential differential games (OPDGs). Potential games are notable for their benefits, such as the computability and guaranteed existence of Nash Equilibria. While previous research has analyzed ordinal potential static games, their applicability to various engineering applications remains limited. Despite the earlier introduction of OPDGs, a systematic method for identifying a potential game for a given LQ differential game has not yet been developed. To address this gap, we propose two identification methods to provide the quadratic potential cost function for a given LQ differential game. Both methods are based on linear matrix inequalities (LMIs). The first method aims to minimize the condition number of the potential cost function's parameters, offering a faster and more precise technique compared to earlier solutions. In addition, we present an evaluation of the feasibility of the structural requirements of the system. The second method, with a less rigid formulation, can identify LQ OPDGs in cases where the first method fails. These novel identification methods are verified through simulations, demonstrating their advantages and potential in designing and analyzing cooperative control systems. [ABSTRACT FROM AUTHOR]

We consider a resource extraction model with the possibility of extending the resource stock by drilling efforts and an oligopolistic competition of symmetric firms. Assuming a hyperbolic market demand, we adopt the open-loop Nash equilibrium to analyze and compare the outcomes of a private vs a common resource pool. For both cases, the drilling efforts of the individual firms strongly depend on the market structure. For a low competition, the efforts increase the number of firms, and the opposite is true for a high competition. On the other hand, aggregate drilling efforts are different among the two types of pools and are opposite to Schumpeterian's hypothesis (i.e., they are either an inverted U-shaped (private pools) or strictly increasing (single pools)).

This study investigates the green development of the South-to-North Water Diversion Project (SNWDP) by exploring the synergistic reduction of water pollution and carbon emissions. Firstly, a differential game model is constructed to reduce pollution and carbon in the water supply chain. Secondly, the emission reduction strategies of water source and receiving areas under centralized and decentralized decision models are compared and analyzed and a cost-sharing contract is designed to coordinate the supply chain. Finally, numerical analysis is used to compare and analyze the important parameters to draw conclusions. The results show that: (1) Collaboration between water source and receiving areas influences emission reduction efforts, with the highest achieved under the centralized model. (2) The cost-sharing contract improves efforts and addresses ‘free-riding’ in the decentralized model. (3) The SNWDP can achieve a win-win situation in terms of both environmental and economic benefits by promoting the synergy of pollution reduction and carbon reduction. HIGHLIGHTS Established a model for synergistic management of low carbon and water pollution control in water supply chain.; Investigated the benefits of low carbon and water pollution control in the water supply chain under various decision models.; Designed a cost-sharing contract to improve synergies in the water supply chain.; Carbon and pollution reduction can improve the economic and environmental benefits of the SNWDP.;

Aiming at the situation of random changes in tactics and uncertain information in the confrontation pattern between missile and target in the air combat environment. By analyzing the motion relationship of a single missile attack to determine the target, the principles of dynamic game and differential game are introduced into the relative motion relationship between missile and target. In the dynamic game process of attack and defense confrontation, the terminal guidance problem of a single missile attacking and intercepting the target is modeled as a "one to one missile target" game model. The missile guidance law based on game theory under mixed strategy and the global strategic situation of bilateral optimization of the opponent at a certain moment are proposed. The method of model rolling prediction combined with differential game theory is introduced to simulate the confrontation between missile and target in uncertain attack and defense environments. The simulation results show that using this method can reduce the miss distance of missile to target and improve the hit accuracy of missile. The model provides a basis for missile attack and defense operations.

The affine formation maneuver (AFM) control problem using the zero-sum differential game is studied for a second-order multi-agent system (MAS) against cyber-injection attacks on the actuators. First and foremost, the distributed control signals and the cyber-injected attack signals are viewed as two competing teams. Then, the leader-follower tracking error system is established where the followers aim to track the AFMs of the leaders. The leader-follower distributed optimal control policies and the cyber-attack attenuation policies are derived from the Nash equilibrium solution of the Hamilton-Jacobi-Isaac (HJI) equation. The solution of the HJI equation is estimated with the aid of the critic neural network. The weights of the critic system are learned online based on the adaptive dynamic programming (ADP) scheme. The uniformly ultimately bounded stability of the closed-loop system under the formulated zero-sum differential game control strategy has been proven by the Lyapunov stability theorem. A simulation example illustrates that the presented control method can attenuate cyber-injection attacks on the actuators and maintain the AFMs of the agents. [ABSTRACT FROM AUTHOR]

This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2. [ABSTRACT FROM AUTHOR]

This article presents an effective computational method based on the orthogonal collocation on finite element for nonlinear pursuit‐evasion differential game problems. The original problems are transformed into two dynamic optimization problems at first, so that the difficulty of obtaining the solution is reduced. To improve the convergence rate and the efficiency, the sensitivities describing the influence of control and interval parameters on state are derived through the discretized dynamic equations for the resulting nonlinear programming problem. The convergence speed is introduced to measure the performance in the upper level iteration. The main structure and the algorithm of the method are also given. Two demonstrative differential game problems with different scenarios from practice are studied. Compared with the approach without sensitivity information, the proposed method needs less function evaluations and saves at least 68.4% of the computational time. The research results show the effectiveness of proposed approach. [ABSTRACT FROM AUTHOR]

We consider the generalized nonstationary Pontryagin example for a game with many players and the same dynamic and inertial capabilities of the players. We obtain sufficient conditions for the multiple capture of a given number of evaders by a group of pursuers, provided that the evaders use program strategies and each of the pursuers captures at most one evader. [ABSTRACT FROM AUTHOR]

Innovative products entering the market will cause dynamic changes in market demand, and consumers' purchase regret and their return behavior make the market environment more and more complex, which in turn affects the dynamic decision-making in the supply chain. In this paper, under the situation of discrete decision time, combining with the objective reality, we make discrete modification to the classical Bass diffusion model (Bass model), construct a manufacturer-led, retailer-followed supply chain differential game model, analyze the optimal decision-making of the manufacturer and the retailer by combining with the theory of discrete optimal control, and then verify the conclusions by numerical simulation. The results show that: when retailers purchase directly from the manufacturer and sell in the market, the optimal pricing of the innovative product can make the supply chain as a whole, realizing Pareto optimality; consumer's purchase regret will increase the amount of returns, which will lead to the decrease of product sales and the profits of the manufacturer and the retailer; when the innovative product accounts for a different share of the market, the impact of purchase regret on the wholesale price and the retail price are also different. Therefore, manufacturers need to have an extensive comprehension of the market to minimize the negative effects of consumer regret and returns, and to formulate a reasonable pricing strategy for their products to gain as much profit as possible. [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of differential graphical games for multiagent systems (MASs) with external disturbances. In particular, the system considered in this article allows the leader to have an unknown bounded control input, and the communication topology between agents is allowed to be a weakly restricted structure with only containing a directed spanning tree. We propose a novel dynamic sliding mode control strategy to dispose the presence of leader's nonzero control input and external disturbances. Furthermore, a nominal controller based on the worst‐case control strategies of neighbors is designed to provide a distributed solution for differential graphics games of MASs. Finally, our algorithms are applied to solve multiple unmanned surface vessels games. Simulation studies are provided to test the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-player zero-sum stochastic differential games with regime switching. The stochastic system is driven by Brownian motion and Markov chains. The cost functionals are defined by controlled backward stochastic differential equations (BSDEs). In this framework, we allow the control processes to depend on the information from the past. By making full use of the construction of Markov chains, we prove that the upper value function and lower value function are indeed deterministic, while the cost functionals are random. We establish the dynamic programming principle (DPP), and obtain that the lower and upper value functions are the unique viscosity solutions of the corresponding systems for partial differential equations (PDEs) of Hamilton-Jacobi-Bellman-Isaacs' (HJBI) type. [ABSTRACT FROM AUTHOR]

With the continuous growth of low-carbon economy and people's increasing preference for low-carbon products, the building carbon emission reduction (CER) has been highly concerned by the government. Therefore, in this study, we establish a game model through differential game to analyze the impact of government policies on building's CER. We get: (1) Compared with the carbon tax policy, the government policy combining carbon tax and cost subsidy is more conducive to reducing carbon emissions; (2) For the three government policies, when the cost subsidy rate is lower than a fixed value, the government policy combining carbon tax and cost subsidy is the optimal strategy for building's CER. When the cost subsidy rate is greater than a fixed value, the cost subsidy policy is the best strategy for building's CER. [ABSTRACT FROM AUTHOR]

Shared manufacturing is a new business form that focuses on all aspects of production and manufacturing, mainly relying on the shared manufacturing platform to achieve the optimal allocation of idle resources. For enterprises, in the process of deciding to lease idle resources, the pricing and advertising investment efficiency of the shared manufacturing platform is a valuable research issue. The shared manufacturing model in this paper consists of one manufacturer and one shared manufacturing platform, which will invest in cooperative advertising while the shared manufacturing process is completed. The cooperative advertising involves four models: the traditional cooperation model, the cost-sharing contract model, the revenue-sharing contract model, and the bilateral cost-sharing contract model. We investigate the impact of some key parameters on the prices and profits of the manufacturer and the shared manufacturing platform based on the differential game. The numerical examples demonstrate the viability of the model. Finally, we provide suggestions based on the decision-making of the manufacturer and the shared manufacturing platform under different cooperative advertising models. [ABSTRACT FROM AUTHOR]

The closed-loop supply chain (CLSC) focusing on product recycling is getting more and more attention. The question of how to motivate CLSC participants to implement product recycling effectively has come into focus. We study the optimal strategy of a two-echelon CLSC under different incentive mechanisms, characterize the return rate in the form of a state equation for its dynamic behavior and construct a closed-loop supply chain dynamic model. Subsequently, we discuss two incentives—the cooperative promotion incentive and wholesale price discount incentive—based on the no-incentive game and investigate the optimal strategy, steady-state return rate, and revenue of each CLSC player in the three scenarios. Then, the choice of incentives by manufacturers and retailers under different scenarios is determined by comparative analysis. The results indicate that retailers always favor the cooperative promotion incentive. For manufacturers, the wholesale price discount incentive is more advantageous only when the recycling efforts outperform retailers' promotion efforts in unit cost. In most other cases, the cooperative promotional incentive is more popular since it will generate more revenue for each participant and the entire supply chain. Moreover, there is a Profit-Pareto-improvement region when the CLSC implements the cooperative promotion incentive. Plain language summary: This research studies a closed-loop supply chain (CLSC) consisting of a manufacturer and a retailer, in which the manufacturer is responsible for product recycling. We characterize the return rate in the form of a state equation and construct the dynamic model of the CLSC, fully considering the dynamic peculiarities of recycling. Then three models based on the no-incentive game, the cooperative promotion incentive and the wholesale price discount incentive are conducted. We also conduct a comparative analysis of the optimal strategy and revenue in three scenarios to further study whether the implementation of these incentives will benefit all CLSC members in terms of increased incomes, improved environmental performance (improved return rates) and enhanced social performance (increased sales). The results show that the cooperative promotion incentive is beneficial to retailers and CLSCs. It can realize the Profit-Pareto improvement and enhance the economic, environmental and social performance of CLSCs. [ABSTRACT FROM AUTHOR]

We consider deterministic mean field games (MFGs) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of first order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFGs with cost functional satisfying various additional assumptions, but not the classical Lasry--Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFGs some results of weak KAM theory. [ABSTRACT FROM AUTHOR]

The problem of approach of controlled objects with different inertia in dynamic game problems is considered. For such controlled objects, it is characteristic that the Pontryagin condition is not satisfied on a time interval. To solve the problem, a special multivalued mapping and a matrix function are introduced to equalize the players' control resources, and then the additional resource is compensated by the body component of the cylindrical terminal set. With the use of the lower resolving function for objects with different inertia, two modified schemes of the first direct Pontryagin method are proposed, which guarantee the successful completion of the conflict-controlled process in the class of countercontrols. The upper resolving function is introduced, and the corresponding modified schemes of the method of resolving functions are presented for controlled objects with different inertia in the class of quasi-strategies and counter-controls. New theoretical results are illustrated by a model example. [ABSTRACT FROM AUTHOR]

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This study aims to solve the current pricing and coordination problems in the fresh produce supply chain by proposing a dynamic pricing and coordination differential game model based on traceability goodwill to optimize the operational efficiency and profitability of the supply chain. Freshness and traceability are critical consumer trust and brand value factors incorporated into the dynamic pricing and coordination model in the agricultural supply chain. The research methodology of this paper integrates critical factors such as supply chain cost and market demand to construct a comprehensive differential game model. It also highlights the motivation of dynamic pricing strategy and coordination mechanism based on traceability goodwill. Through traceability goodwill, operating companies can better assess and manage product quality and improve consumer trust in agricultural products, thus enhancing brand value. The dynamic pricing strategy based on traceability goodwill can flexibly adjust prices according to changes in the quality characteristics of agricultural products and market demand, realizing a dynamic balance between supply and demand. On the other hand, the mutual cost-sharing and coordination mechanism promotes the synergy between the various links of the supply chain. It improves overall efficiency through cooperation coo, coordination, and other means. The validity and reliability of the model are verified through numerical simulation experiments. The numerical simulation results show that the dynamic pricing and coordination model based on traceability will significantly improve supply chain efficiency and profitability compared with the traditional model. Specifically, the interests of supply chain participants are balanced, and the overall supply chain operational efficiency is improved. The main contribution of this study is to propose a dynamic pricing and coordination model based on traceability goodwill and verify its superiority through numerical simulation experiments. This study provides a valuable reference for fresh agricultural products supply chain management and offers new ideas for improving supply chain efficiency and profitability. [ABSTRACT FROM AUTHOR]

In this paper, we study the multi‐pursuer single‐evader pursuit‐evasion (MSPE) differential game in a continuous environment with the consideration of obstacles. We propose a novel pursuit‐evasion algorithm based on reinforcement learning and transfer learning. In the source task learning stage, we employ the Q‐learning and value function approximation method to overcome the challenges posed by the large‐scale storage space required by the conventional Q‐table solution method. This approach expands the discrete space to the continuous space by value function approximation and effectively reduces the demand for storage space. During the target task learning stage, we utilize the Gaussian mixture model (GMM) to classify the source tasks. The source policies whose corresponding state‐value sets have the highest probability densities are assigned for the agent in the target task for learning. This methodology not only effectively avoids negative transfer but also enhances the algorithm's generalization ability and convergence speed. Through simulation and experiment, we demonstrate the algorithm's effectiveness. [ABSTRACT FROM AUTHOR]

This paper addresses the active target protection problem in a three-player (Target–Attacker–Defender, TAD) conflict by proposing a cooperative integrated guidance and control (IGC) strategy. Unlike previous studies that have designed guidance and control loops separately, this work establishes an IGC model by linearizing both the translational motion and the rotational motion of the vehicles, thereby generating actuator commands directly. This model integrates the kinematics and short-period dynamics, providing a more comprehensive and accurate representation of the vehicles' characteristics. Based on the linearization and order reduction, differential game theory and the sweep method are employed to derive and analytically solve the Riccati differential equation, yielding an optimal control strategy with an explicit expression. The theoretical rigor of the proposed approach is ensured through a proof of optimality sufficiency. Furthermore, factors influencing the computational accuracy of the Riccati equation solution, including the singular values of the control matrix and condition numbers of the solution matrix, are analyzed. Taking into account the dynamic response and limitations of the actuators, numerical simulations demonstrate the effectiveness and superiority of the proposed IGC strategy in intercepting the attacker and protecting the target compared to traditional separated guidance and control designs. [ABSTRACT FROM AUTHOR]

With the rise of influential customers and environmental issues, this paper studies the alliance effect of the manufacturer and its weak customers by differential Stackelberg game. Low-carbon production and dynamic pricing strategies are analyzed in alliance and no-alliance situations. The results find that forming an alliance between the manufacturer and its weaker customer helps them resist the advantages of a stronger retailer. Compared to the no-alliance strategy, the alliance strategy will decrease their profits. The stronger customer needs to set a lower retail price toward the alliance between the manufacturer and the weak customer. And it can narrow the price gap in two situations by increasing its proportion of sales revenue. Moreover, when consumers care less about the product's environmental problems, all the partner firms in the supply chain will price lower and make fewer efforts on low-carbon production. [ABSTRACT FROM AUTHOR]

To achieve the intelligent interception of different types of maneuvering evaders, based on deep reinforcement learning, a novel intelligent differential game guidance law is proposed in the continuous action domain. Different from traditional guidance laws, the proposed guidance law can avoid tedious manual settings and save cost efforts. First, the interception problem is transformed into the pursuit–evasion game problem, which is solved by zero-sum differential game theory. Next, the Nash equilibrium strategy is obtained through the Markov game process. To implement the proposed intelligent differential game guidance law, an actor–critic neural network based on deep deterministic policy gradient is constructed to calculate the saddle point of the differential game guidance problem. Then, a reward function is designed, which includes the tradeoffs among guidance accuracy, energy consumption, and interception time. Finally, compared with traditional methods, the interception accuracy of the proposed intelligent differential game guidance law is 99.2%, energy consumption is reduced by 47%, and simulation time is shortened by 1.58 s. All results reveal that the proposed intelligent differential game guidance law has better intelligent decision-making ability. [ABSTRACT FROM AUTHOR]

In competitive scenarios, we analyze the optimal pricing, launch timing, and influencing factors of new products by establishing three differential game models. The findings indicate that (1) the depreciation rate of old products directly impacts the pricing strategies of advantaged and disadvantaged enterprises. Enterprises with competitive advantages tend to set higher prices for new products when old products depreciate faster, while those with fewer advantages adjust their pricing strategies by initially increasing prices for the latest prices and eventually reducing them. (2) The rate of old‐product depreciation also affects the timing of new‐product launches. Advantaged enterprises tend to delay their product launches when the impact of the new product on the old ones is significant, while disadvantaged enterprises opt for earlier launches under similar circumstances. (3) The technological level of the new product influences the timing of its launch for both advantaged and disadvantaged enterprises. Higher technological levels prompt advantaged enterprises to launch earlier and disadvantaged enterprises tend to follow suit. (4) High levels of innovation in new products by advantaged enterprises prompt earlier launches by disadvantaged enterprises. These results may offer valuable insights for enterprises in formulating effective product rollover strategies. [ABSTRACT FROM AUTHOR]

We consider a resource extraction model with the possibility of extending the resource stock by drilling efforts and an oligopolistic competition of symmetric firms. Assuming a hyperbolic market demand, we adopt the open-loop Nash equilibrium to analyze and compare the outcomes of a private vs a common resource pool. For both cases, the drilling efforts of the individual firms strongly depend on the market structure. For a low competition, the efforts increase the number of firms, and the opposite is true for a high competition. On the other hand, aggregate drilling efforts are different among the two types of pools and are opposite to Schumpeterian's hypothesis (i.e., they are either an inverted U-shaped (private pools) or strictly increasing (single pools)). [ABSTRACT FROM AUTHOR]

In this foreword to the Special Issue "Nonlinear dynamics and game-theoretical modeling in economics and finance" we review the contributions in the issue highlighting the economic results and the connections with aspects of dynamic analysis. Indeed, the common theme of the contributions is the focus on system dynamics and the latest analytical techniques. This issue is devoted to celebrating the 70th birthday of Professor Laura Gardini, who inspired a generation of scholars in the study of discrete-time systems and global methods of analysis. [ABSTRACT FROM AUTHOR]