Non-cooperative Game Research Articles

Price Stickiness in a Dual‐Channel Supply Chain

ABSTRACTIn this paper, we investigate price stickiness in a dual‐channel supply chain where a manufacturer sells its product directly through an online retailer and indirectly through an offline retailer. We construct a noncooperative game where the manufacturer and the offline retailer decide whether or not to costlessly adjust their prices after a demand shock. If the demand shock is positive, then no price can be sticky at the unique Nash equilibrium. If the demand shock is negative, then we additionally observe equilibria where some or all prices can be sticky. We also show that no equilibrium is always Pareto optimal.

Non-cooperative game optimization for virtual power plants considering carbon trading market

Non-cooperative game optimization for virtual power plants considering carbon trading market

Centralized and Decentralized Event-Triggered Nash Equilibrium-Seeking Strategies for Heterogeneous Multi-Agent Systems

This paper addresses the event-triggered Nash equilibrium-seeking problem for non-cooperative games played by heterogeneous multi-agent systems. Unlike homogeneous multi-agent systems, heterogeneous multi-agent systems consist of agents with different dynamic structures, making it difficult to design control schemes and construct event-triggering conditions for such systems. In this paper, a novel centralized event-triggered Nash equilibrium-seeking strategy and a novel decentralized event-triggered Nash equilibrium-seeking strategy are proposed. The corresponding centralized and decentralized event-triggering conditions are derived. The convergence properties of the proposed centralized and decentralized strategies are proved. Further theoretical analyses illustrate that Zeno behavior does not exist under the proposed strategies. Finally, the effectiveness and efficiency of both centralized and decentralized strategies are presented through numerical experiments. The experimental results illustrate that under both strategies, heterogeneous multi-agent systems achieve the Nash equilibrium successfully, and the communication consumption among agents is significantly reduced.

Distributed nash equilibrium seeking for heterogeneous second-order nonlinear noncooperative games with communication delays

This paper investigates the noncooperative game problems over weight-balanced digraphs with communication delays. Compared with existing noncooperative games, a class of noncooperative games that are more aligned with practical systems is discussed, where all agents have uncertain heterogeneous second-order nonlinear dynamics. Then, considering two different types of communication delays, respectively slowly-varying communication delays and fast-varying communication delays, the distributed algorithm is designed to seek the Nash equilibrium (NE). By designing appropriate Lyapunov functions and constructing linear matrix inequalities, the designed algorithm has been proven to be effective, even though second-order nonlinear dynamics and communication delays increase the challenges of system convergence analysis. Furthermore, simulation cases are given to illustrate the validity of the designed algorithm.

Modelling diversity in hospital strategies in city-scale ambulance dispatching with coupled game-theoretic model and discrete-event simulation.

The optimization in the ambulance dispatching process is significant for patients who need early treatments. However, the problem of dynamic ambulance redeployment for destination hospital selection has rarely been investigated. The paper proposes an approach to model and simulate the ambulance dispatching process in multi-agent healthcare environments of large cities. The proposed approach is based on using the coupled game-theoretic (GT) approach to identify hospital strategies (considering hospitals as players within a non-cooperative game) and performing discrete-event simulation (DES) of patient delivery and provision of healthcare services to evaluate ambulance dispatching (selection of target hospital). Assuming the collective nature of decisions on patient delivery, the approach assesses the influence of the diverse behaviors of hospitals on system performance with possible further optimization of this performance. The approach is studied through a series of cases starting with a simplified 1D model and proceeding with a coupled 2D model and real-world application. The study considers the problem of dispatching ambulances to patients with the Acute Coronary Syndrome (ACS) directed to the Percutaneous Coronary Intervention (PCI) in the target hospital. A real-world case study of data from Saint Petersburg (Russia) is analyzed showing the better conformity of the global characteristics (mortality rate) of the healthcare system with the proposed approach being applied to discovering the agents' diverse behavior.

Cooperation strategy of government-enterprise supply chain finance based on differential game

PurposeThe government plays an important role in the financing process of small and medium enterprises (SMEs), but the current government-enterprise cooperation (GEC) mechanism cannot well solve the financing problem of SMEs. In addition, since government-enterprise cooperation is a long-term dynamic process, this study aims to explore the cooperation strategy between the government and core enterprises in supply chain finance (SCF) under the dynamic structure.Design/methodology/approachConsidering both parties have the characteristics of disappointment aversion, our research constructs a game theory model of government subsidy and the effort of the enterprise to implement SCF based on differential game and studies different game strategies in the non-cooperative game, the Stackelberg game and the cooperative game.FindingsOur findings show that the government subsidy can significantly spur the enterprise in the supply chain to implement SCF. We also find that the limitation of the government subsidy exists. In addition, the optimal strategy, the optimal benefit and the total benefit of the financing system formed by the government and the enterprise in the cooperative game are better than those in the non-cooperative game. Pareto optimality is achieved.Research limitations/implicationsThe limitations of this paper are: (1) In theory, this paper only takes the government and enterprise as game subjects without considering other participants in the supply chain. In addition, only disappointment aversion is taken into consideration while in reality the participants often exhibit multiple behavior preferences. (2) In methodology, only the numerical solution is given through the solution algorithm and all parameters are assumed to be determined as time changes for the convenience of calculation.Practical implicationsThe government can motivate enterprises in the supply chain to implement SCF by providing subsidies. However, it should be noted that excessive subsidy will make the enterprise dependent on the government and as a result, decrease the effort level to implement SCF. Thus, it is necessary for the government to keep track of the business conditions of the enterprise to make a subsidy strategy. In addition, the government can reduce the impact of disappointing aversion by making more targeted policies and taking risk management measures.Social implicationsOn one hand, it is necessary for the government to keep track of the business conditions of the enterprise to make a subsidy strategy. In addition, the government can reduce the impact of their own disappointing aversion by making more targeted policies and adopting risk management measures. On the other, it is important for the enterprise to improve the regulatory mechanism and optimize the compensation structure of decision-makers to inhibit the impact of decision-makers' disappointing aversion.Originality/valueThis study is the first to investigate the mechanism of GEC in the setting of SCF based on a differential game. Furthermore, our study provides a theoretical basis for the government and enterprises to cooperate in SCF.

A Class of Pursuit Problems in 3D Space via Noncooperative Stochastic Differential Games

This paper investigates three-dimensional pursuit problems in noncooperative stochastic differential games. By introducing a novel polynomial value function capable of addressing high-dimensional dynamic systems, the forward–backward stochastic differential equations (FBSDEs) for optimal strategies are derived. The uniqueness of the value function under bounded control inputs is rigorously established as a theoretical foundation. The proposed methodology constructs optimal closed-loop feedback strategies for both pursuers and evaders, ensuring state convergence and solution uniqueness. Furthermore, the Lebesgue measure of the barrier surface is computed, enabling the design of strategies for scenarios involving multiple pursuers and evaders. To validate its applicability, the method is applied to missile interception games. Simulations confirm that the optimal strategies enable pursuers to consistently intercept evaders under stochastic dynamics, demonstrating the robustness and practical relevance of the approach in pursuit–evasion problems.

Cybersecurity investments and cyber insurance purchases in a non-cooperative game

Abstract The growing concern over cyber risk has become a pivotal issue in the business world. Firms can mitigate this risk through two primary strategies: investing in cybersecurity practices and purchasing cyber insurance. Cybersecurity investments reduce the compromise probability, while cyber insurance transfers potential losses to insurers. This study employs a network model for the spread of infection among interconnected firms and investigates how each firm’s decisions impact each other. We analyze a non-cooperative game in which each firm aims to optimize its objective function through choices of cybersecurity level and insurance coverage ratio. We find that each firm’s cybersecurity investment and insurance purchase are strategic complements. Within this game, we derive sufficient conditions for the existence and uniqueness of Nash equilibrium and demonstrate its inefficiency. These theoretical results form the foundation for our numerical studies, allowing us compute firms’ equilibrium decisions on cybersecurity investments and insurance purchases across various network structures. The numerical results shed light on the impact of network structure on equilibrium decisions and explore how varying insurance premiums influence firms’ cybersecurity investments.

GTD‐CER: Game‐Theoretic Dynamic Clustering for Enhanced Efficiency and Reliability in Wireless Sensor Networks

ABSTRACTEfficient cluster formation is a significant issue in wireless sensor networks (WSNs) for optimizing energy consumption, enhancing data reliability, and prolonging network lifetime. This paper proposes a dynamic cluster formation scheme modeled as a noncooperative game, where each sensor node acts as a player making strategic decisions on whether to become a cluster head (CH) or remain a member node (MN). In this role selection decision‐making phase, nodes evaluate their potential utility function, which incorporates energy consumption, data reliability, and data forwarding efficiency. Nodes dynamically switch roles based on their utility to achieve an optimal network configuration. Simulations conducted using the NS‐3 network simulator demonstrate that the proposed game theory‐based clustering scheme significantly improves network performance. We analyze key measures like energy consumption, packet delivery ratio (PDR), and data forwarding, and the results shows that the proposed scheme reduces energy consumption by 25%, improves PDR by 15%, and extends the network lifetime by 20%, compared to traditional methods. The dynamic role‐switching mechanism prolongs network lifetime by preventing the rapid depletion of node energy, ensuring stable and efficient network operations. It significantly enhances network performance, stability, and longevity.

Non-cooperative inventory games for defective items and quantity discounts using strategic complementarity

In this paper, the mathematical analysis for obtaining the equilibrium of the inventory games under strategic complementarities, the existence of defective items, and quantity discounts have been analyzed. The inventory system consists of many buyers who order a single type of product from one supplier. They compete with each other as a player in a non-cooperative game with strategic complementary. They maximize a supermodular payoff function and take into account some fraction of defective items from a lot of the arrival products. The concept of supermodular games is used to obtain the equilibria of these problems. A new existence theorem of Nash equilibrium in a specific condition has been proved. The optimum analysis has been justified for two conditions, that is the condition without discount and another without it. The numerical computations are provided using Python programming. At the end, the numerical result shows that elements of the Nash equilibrium set can be altered when discounts are considered. A quantity discount policy can be used by the supplier to prevent players from choosing the least Nash equilibrium.

Physical layer security in double RIS-aided WPCN systems based on non-cooperative game

Physical layer security in double RIS-aided WPCN systems based on non-cooperative game

Optimization Method of Guide Roller Cloud Manufacturing Service Composition based on Non-Cooperative Game

Optimization Method of Guide Roller Cloud Manufacturing Service Composition based on Non-Cooperative Game

Pure Nash equilibria in weighted matroid congestion games with non-additive aggregation and beyond

Congestion games offer a primary model in the study of pure Nash equilibria in non-cooperative games, and a number of generalized models have been proposed in the literature. One line of generalization includes weighted congestion games, in which the cost of a resource is a function of the total weight of the players choosing that resource. Another line includes congestion games with mixed costs, in which the cost imposed on a player is a convex combination of the total cost and the maximum cost of the resources in her strategy. This model is further generalized to that of congestion games with non-additive aggregation. For the above models, the existence of a pure Nash equilibrium is proved under some assumptions, including the case in which the strategy space of each player is the base family of a matroid and the case in which the cost functions have a certain kind of monotonicity. In this paper, we deal with common generalizations of these two lines, namely weighted matroid congestion games with non-additive aggregation, and its further generalization. Our main technical contribution is a proof of the existence of pure Nash equilibria in these generalized models under a simplified assumption on the monotonicity, which provides a common extension of the previous results. We also present an extension on the existence of pure Nash equilibria in weighted matroid congestion games with mixed costs.

Study on equilibrium strategies for transboundary pollution under competitive conditions

Competitive relationships among players plays an increasing role on transboundary pollution management. In this paper, a transboundary pollution game between two competitive regions as a player is constructed to explore the equilibrium strategies of output and pollution abatement efforts. The dynamic of the pollution stocks for regions are modeled separately in terms of a differential equation. Subsequently, the corresponding equilibrium strategies and value functions are derived in two regimes: Nash non-cooperative and Stackelberg leader–follower games. And the expectation and variances of pollution stocks are calculated over time. Our results are summarized as follows. First, an increase in the proportion of transboundary pollution increases the output of pollution upstream region, decreases the output of pollution downstream region, and ultimately raises the aggregate pollution stock of both regions. Second, regional competitive preference can enhance the local abatement effort and reduce the aggregate pollution stock. Finally, the total social welfare and environmental quality are higher in the Nash non-cooperative game, but pollution downstream region seek to be a leader in the Stackelberg game.

Triangles of conflicts and hexamatrix games

The paper addresses one class of finite non-cooperative games (with finite numbers of strategies for each player) – Polymatrix Games of E.B. Yanovskaya. Namely, we study in detail the 3-players Polymatrix Games, the so-called Hexamatrix Games (HMG), which can be completely described by six matrices. A number of model examples of 3-sides real-life conflicts are presented and formulated as HMG. The possibility of using hexamatrix games to model economic relationships between three participants is demonstrated. To find a Nash Equilibrium in the formulated games, we use an optimization approach, when the problem is transformed into a nonconvex optimization problem with a bilinear structure. The latter is solved by the A.S. Strekalovsky’s Global Search Theory (GST) for (d.c.) optimization problems with objective functions represented as the difference of two convex functions.

Research on Decision-Making and Coordination of Closed-Loop Supply Chain Under Green and Low-Carbon Perspective

This study explores low-carbon tech investment decisions in a closed-loop supply chain and their impact on supply chain optimization and relationships, using non-cooperative and cooperative game theories. This study examines the behavior patterns and potential gains of suppliers and manufacturers in independent vs. cooperative circumstances using model simulations. This study reveals that the following: (1) Optimal pricing, technology investment, profit, and cost-sharing ratios for low-carbon technologies can be determined under certain conditions to maximize stakeholder profits. (2) Suppliers and manufacturers can mutually benefit in supplier-driven markets. (3) Manufacturers have greater investment opportunities in low-carbon technology when suppliers have market power. (4) Provider utility decreases as consumer preference for low-carbon products rises. (5) The dominant market structure for low-carbon products evolves with the market stage. These findings provide strategic guidance for green supply chain management. The research presented in this paper offers a theoretical foundation and strategic direction for low-carbon technology investment in CLSCs.

A game-theoretic driver steering model with individual risk perception field generation

Driver-automation shared steering control (SSC) has emerged as a promising technology for enhancing vehicle safety, but desire to achieve seamless collaboration between the driver and automation requires an in-depth understanding of driver steering behavior in interaction with automation. In this paper, we introduce a game-theoretic driver steering model with individual risk perception field generation. Firstly, a driver risk perception field is developed based on a novel concept of potential injury risk (PIR) to provide a quantitative estimation of the driver’s perceived driving risk. This approach offers an explicit and physically meaningful structure for simulating the driver’s risk perception process and elucidating the reasons for discrepancies in risk perception. Then, this driver risk perception field is integrated into the framework of non-cooperative Nash game to model the steering interaction between the driver and automation, and the analytical expression of Nash equilibrium is derived in detail. The resulting combined driver model effectively captures the driver adaptation at both the control and planning levels. Next, the key parameters of the combined driver model and its comparators are identified using measured driver steering behavior data from thirty subjects in a series of driving simulator experiments. Finally, the effectiveness and superiority of the combined driver model is validated through a comprehensive comparative analysis. The results demonstrate that the combined driver model achieves the lowest prediction errors compared to its comparators and effectively captures the individual differences in risk perception and steering behavior.

Battery swapping scheduling for electric vehicles: a non-cooperative game approach

In recent years, electric vehicle (EV) battery-swapping technology has rapidly evolved and is expected to become widely prevalent shortly. Therefore, it is crucial to develop efficient battery-swapping scheduling algorithms to optimize the operations of battery-swapping systems. This paper proposes a non-cooperative game approach for the battery-swapping scheduling of EVs. To reduce the waiting time for battery swapping and improve the scheduling efficiency of EVs, a swapping process model inspired by the job-shop scheduling problem is proposed, and the cost function of each EV comprehensively considers the travel time, waiting time, and battery swapping price. To capture the competitive relationship among EVs, a non-cooperative game model for battery swapping scheduling is established considering the finite quantities of batteries and swapping grippers. To find the pure strategy Nash equilibrium, an iterative best response algorithm is developed, satisfying constraints including those couple decisions of different EVs. Case studies demonstrate the fairness and scheduling efficiency of the proposed approach.

A Machine Learning Method for Stackelberg Mean Field Games

We propose a single-level numerical approach to solve Stackelberg mean field game (MFG) problems. In the Stackelberg MFG, an infinite population of agents plays a noncooperative game and chooses their controls to optimize their individual objectives while interacting with the principal and other agents through the population distribution. The principal can influence the mean field Nash equilibrium at the population level through policies, and she optimizes her own objective, which depends on the population distribution. This leads to a bilevel problem between the principal and mean field of agents that cannot be solved using traditional methods for MFGs. We propose a reformulation of this problem as a single-level mean field optimal control problem through a penalization approach. We prove convergence of the reformulated problem to the original problem. We propose a machine learning method based on (feed-forward and recurrent) neural networks and illustrate it on several examples from the literature. We also give a modified example to show the scalability of the proposed method and show its efficiency by comparing its performance with an alternative bilevel approach.

Distributed NE and GNE seeking for heterogeneous multi-player games over directed communication networks.

The non-cooperation game with heterogeneous dynamics is of both theoretical significance and practical relevance because of its extensive penetration into various fields, such as the game confrontation composed of unmanned aerial vehicles and unmanned vehicles or the power generation systems with varied turbine assemblies. To solve such a game problem, this paper investigates distributed Nash equilibrium (NE) and generalized Nash equilibrium (GNE) seeking problems for heterogeneous multi-player systems in non-cooperation games. First, by incorporating the output regulation technique, a distributed NE seeking strategy is designed for heterogeneous multi-player games over undirected communication networks. Then, the GNE seeking problem subject to coupling constraints among multiple players is further studied and a distributed GNE seeking strategy is developed by utilizing the Lagrange multiplier and the output regulation technique. Second, in the case of directed communication networks, both distributed NE and GNE seeking strategies are proposed by employing new type of auxiliary variables to compensate for the unbalanced networks. Compared with some existing results on distributed NE and GNE seeking strategies, both the dynamics of players and communication networks are extended further. Finally, numerical simulations are presented to show the effectiveness of the NE and GNE seeking strategies.