Strategy Nash Equilibrium Research Articles

Equilibrium Control in Uncertain Linear Quadratic Differential Games with V-Jumps and State Delays: A Case Study on Carbon Emission Reduction

Uncertainty, time delays, and jumps often coexist in dynamic game problems due to the complexity of the environment. To address such issues, we can utilize uncertain delay differential equations with jumps to depict the dynamic changes in differential game problems that involve uncertain noise, delays, and jumps. In this paper, we first examine a linear quadratic differential game optimistic value problem within an uncertain environment characterized by jumps and delays. By applying the Z(x,y) transform, we convert the infinite-dimensional problem into a finite-dimensional one. We then demonstrate that the condition for the existence of a Nash equilibrium strategy is equivalent to the existence of solutions to two cross-coupled matrix Riccati equations. Furthermore, we explore the saddle point equilibrium strategy of the linear quadratic differential game optimistic value model and derive the corresponding saddle point equilibrium solution. Finally, we apply our results to solve a carbon emission reduction game problem.

Constructive Approximate Nash Equilibrium for In-Orbit Target Enclosing with Collision Avoidance and Full-state Constraint via Nonzero-Sum Differential Games

The problem of in-orbit cooperative target enclosing involving N thrust-limited satellites under collision avoidance and maneuver amplitude constraints is studied. In order to find global optimal trajectories for target enclosing task with all constraints above, by integrating the collision threat and maneuver boundaries into a novel nonlinear cost functional, the studied target enclosing problem is described as a nonlinear nonzero-sum differential game. Further, to avoid iterative calculations caused by traditional nonlinear-game-solving methods, an approximate solution which can be constructed directly is designed. Then the approximate Nash equilibrium strategies can be educed by the constructive approximate solution for the proposed nonzero-sum game. Analysis shows that the proposed control strategies can asymptotically approach the exact one and can ensure a zero-error tracking of the enclosing configuration. Simulation results illustrate lower time costs and better enclosing accuracy while the collision avoidance and maneuver amplitude constraints are satisfied simultaneously.

Blockchain-Enabled Secure and Authentic Nash Equilibrium Strategies for Heterogeneous Networked Hub of Electric Vehicle Charging Stations

In the networked enlarged electric vehicle (EV) charging infrastructures, the security and authenticity of the stakeholders involved in the EV energy market pool are prime important. This paper proposes an EV network hub (EVNH) comprising electric vehicles, electric vehicle aggregators (EVAs), and charging nodes in the networked EV energy market pool. The various EVAs implement different heterogeneous blockchains. The EV network hub facilitates blockchain-based secure and resilient energy trading under the grid to vehicle and vehicle to grid. The paper emphasizes interoperability challenges involving different blockchains to communicate and transfer assets or data between them. We suggest secure and trustworthy energy trading across various EVAs using multiple EV tokens for EV energy trading through cross-chain communications. The EVAs consider a Nash equilibrium-seeking strategy to find the Nash equilibrium in the non-cooperative game of EVAs. The effectiveness of the proposed EV network hub is tested using MATLAB, Solidity, and Python software.

Service pricing and charging strategy for video platforms considering consumer preferences

AbstractHeterogeneity of consumer preferences is a crucial feature of the platform economy, and service price and charging strategies crucially influence customer choice behavior. To enhance consumer experience and boost product value, video platforms offer value‐added services, also known as premium membership features. For instance, Youku's benefits of value‐added service include seamless video playback, high‐definition picture quality, and exclusive episodes. This study constructs a two‐stage Hotelling model from monopoly to duopoly competition. Considering the heterogeneity of consumer preferences for product quality and platform usage costs, we analyze the impact mechanism of the customer preferences heterogeneity and the service pricing of video platform enterprises. In addition, we analyze the Nash equilibrium strategies of competing enterprises in the competition scenario, and we consider two charging strategies: freemium and bundled sales. The results indicate that the consumer sensitivity factors significantly impact the optimal pricing of services and the profits of enterprises and that this impact varies with the heterogeneity of customer composition and the type of charging strategies. For the optimal solutions in a competition scenario, the proportion of quality‐sensitive customers is positive for the freemium model, whereas it is negative for the bundled sales model. Furthermore, we analyze the impact of heterogeneous customer composition on the choice of charging strategy utilizing numerical analysis. This study provides a theoretical basis and decision support for product development and value‐added service pricing that can be applied to video platform enterprises and other media platforms.

Deep Reinforcement Learning-Based Differential Game Guidance Law against Maneuvering Evaders

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.

The Evolution of Firm Connections in Networks: A Differential Game Approach

Optimal control theory can be employed to gain novel insights on the self-organization and structure of networks. We develop a Cournot differential game to analyse the evolutionary dynamics of firm connections within a network. We determine the feedback (Markovian) Nash equilibrium strategies and the steady state of the model and identify the key factors affecting the strategic choice of network firms. Our model confirms the empirical evidence that network firms tend to increase their own connections over time; moreover, such growing connections are mainly affected by the market size and the spillover rate.

The Partial Solution of First Price Sealed Bid Auction in the Scope of Mechanism Design Theory and With Uniformly Distributed Types

In the article the first price sealed bid auction has been analysed within the scope of mechanism design theory and with uniformly distributed types. The sealed bid auction consisting of two of agents has been observed finding the partial solution to the auction. Bayesian Nash equilibrium strategies, the equilibrium state, and the implemented social choice function for the auction have been evaluated. The mechanism has been designed to truthfully implement the social choice function for the auction consisting of two agents. The variables such as the probability of an agent to win, the expected utility of an agent and the expected revenue of the seller have been evaluated.

Social welfare maximization with efficient energy management of community microgrid considering customer behavioral response using MDCLPIS

A novel residential demand response (DR) program is proposed to encourage microgrid customers to participate in demand reduction using a house energy consumption scheduler (HECS) to maximize social welfare by optimally reducing the cost of electricity and to receive maximum incentives. In this research, both Price-based and incentive-based DR programs are considered, and the demand-price elasticity approach is implemented to model them. To make the DR model more effective, customer behavioral response is taken into account, by leveling the customers according to their price elasticity. An optimized non-cooperative energy game theory framework and a Nash equilibrium strategy are employed to model customer response under a multi-time of use (TOU) pricing and incentive scheme, with the aim of achieving the best scheduling of home-controlled appliances while preserving customers’ privacy. This approach maximizes social welfare by minimizing the electricity consumption cost for customers while simultaneously saving on generation costs. The negative externality effect on price rate is internalized between the customers of different consumption levels using multi-dynamic consumption-level pricing and incentive scheme (MDCLPIS). The suggested model distinguishes between consumers’ behavior in response to changes in energy prices and their behavior in response to changes in incentives. Finally, numerical and simulation results demonstrate the suitability and effectiveness of the proposed model and its advantages in terms of customer saved cost by up to 21.26%, microgrid saved cost by up to 29.44%, and getting benefits by optimal scheduling of their load.

Effects of green supplier integration and development in competing supply chains

AbstractThis work develops a Cournot game model to study two competing supply chains, each consisting of one manufacturer and one supplier, producing and selling substitutable green products. The manufacturers decide whether to adopt the green supplier integration (GSI) strategy and the green supplier development (GSD) investment level to improve the product greenness degree. The interaction between the GSD decisions and the GSI strategies are analyzed with different manufacturer GSD capacities and market greenness sensitivity. The results indicate that the manufacturer with a higher GSD capacity will always adopt the GSI strategy to maximize GSD investment level or profit, and the manufacturer with a lower GSD capacity decides whether to adopt the GSI strategy by comparing the GSD capacities of the two manufacturers. The game model is extended from dedicated sourcing to diversified sourcing and hybrid sourcing to obtain optimal sourcing options jointly reached by the two manufacturers. Under these extensions, the GSI strategy always leads to higher GSD investment levels, and adopting the GSI strategies by both manufacturers becomes the only Nash equilibrium strategy combination. Manufacturers choose different sourcing options by judging the difference in their GSD capacities under different criteria, and choose hybrid sourcing when their GSD capacity gap is sufficiently large in a weakly competitive market.

Distributed Task Offloading and Resource Purchasing in NOMA-Enabled Mobile Edge Computing: Hierarchical Game Theoretical Approaches

As the computing resources and the battery capacity of mobile devices are usually limited, it is a feasible solution to offload the computation-intensive tasks generated by mobile devices to edge servers (ESs) in mobile edge computing (MEC) . In this article, we study the multi-user multi-server task offloading problem in MEC systems, where all users compete for the limited communication resources and computing resources. We formulate the offloading problem with the goal of minimizing the cost of the users and maximizing the profits of the ESs. We propose a hierarchical EETORP (Economic and Efficient Task Offloading and Resource Purchasing) framework that includes a two-stage joint optimization process. Then we prove that the problem is NP-complete. For the first stage, we formulate the offloading problem as a multi-channel access game (MCA-Game) and prove theoretically the existence of at least one Nash equilibrium strategy in MCA-Game. Next, we propose a game-based multi-channel access (GMCA) algorithm to obtain the Nash equilibrium strategy and analyze the performance guarantee of the obtained offloading strategy in the worst case. For the second stage, we model the computing resource allocation between the users and ESs by Stackelberg game theory, and reformulate the problem as a resource pricing and purchasing game (PAP-Game). We prove theoretically the property of incentive compatibility and the existence of Stackelberg equilibrium. A game-based pricing and purchasing (GPAP) algorithm is proposed. Finally, a series of both parameter analysis and comparison experiments are carried out, which validate the convergence and effectiveness of the GMCA algorithm and GPAP algorithm.

Game-theoretic modeling and analysis of cyberbullying spreading on OSNs

The rapid development of online social networks (OSNs) has enabled seamless cyberbullying activities, which not only cause psychological harm to victims but also bring disharmony to society. From the perspective of OSN platform administrators, strategic decision making towards when/how to launch an anti-cyberbullying campaign is the key to mitigating its negative impact. Hence, this paper develops a dynamic anti-cyberbullying investment solution to mitigate the adverse impact of strategic cyberbullying phenomena cost-effectively. Specifically, this paper first quantifies the payoffs of the attacker/bullying instigator and the defender/platform administrator based on a newly proposed propagation model. Then, considering the budget-constrained factor for both sides, a differential game-theoretical model is proposed to simulate each party's optimal trade-off between cost and benefit. Next, by utilizing Pontryagin Maximum/Minimum Principle, a set of necessary conditions for a Nash equilibrium strategy profile have been proposed. Based on the forward-backward sweep method, an iterative algorithm (SCR optimization algorithm) is tailored to solve the conditions numerically. Through experiments conducted under Facebook, Youtube and Twitter sub-networks, the algorithm's performance has been evaluated by inspecting its convergence speed and payoff compared with randomized profiles. Consequently, the resulting strategies are verified to exhibit superior performance in dealing with cyberbullying spreading.

Game-Based Approximate Optimal Motion Planning for Safe Human-Swarm Interaction.

Safety as a fundamental requirement for human-swarm interaction has attracted a lot of attention in recent years. Most existing approaches solve a constrained optimization problem at each time step, which has a high real-time requirement. To deal with this challenge, this article formulates the safe human-swarm interaction problem as a Stackerberg-Nash game, in which the optimization is performed over the entire time domain. The leader robot is supposed to be in a dominant position, interacting directly with the human operator to realize trajectory tracking and responsible for guiding the swarm to avoid obstacles. The follower robots always take their best responses to leader's behavior with the purpose of achieving the desired formation. Following the bottom-up principle, we first design the best-response controllers, that is, Nash equilibrium strategies, for the followers. Then, a Lyapunov-like control barrier function-based safety controller and a learning-based formation tracking controller for the leader are designed to realize safe and robust cooperation. We show that the designed controllers can make the robotic swarms move in a desired geometric formation following the human command and modify their motion trajectories autonomously when the human command is unsafe. The effectiveness of the proposed approach is verified through simulation and experiments. The experiment results further show that safety can still be guaranteed even when there exists a dynamic obstacle.

Computing Bayes–Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Determining equilibria in auction games is computationally hard in general, and no exact solution theory is known. We introduce an algorithmic framework in which we discretize type and action space and then learn distributional strategies via online optimization algorithms. We show that the equilibrium of the discretized auction game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the approach approximates the analytical (pure) Bayes–Nash equilibrium closely. In standard models in which agents are symmetric, we find equilibrium in seconds. The method allows for interdependent valuations and different types of utility functions, and it can be used to find equilibrium in auction markets and beyond.

An efficient model‐free adaptive optimal control of continuous‐time nonlinear non‐zero‐sum games based on integral reinforcement learning with exploration

AbstractTo reduce the learning time and space occupation, this study presents a novel model‐free algorithm for obtaining the Nash equilibrium solution of continuous‐time nonlinear non‐zero‐sum games. Based on the integral reinforcement learning method, a new integral HJ equation that can quickly and cooperatively determine the Nash equilibrium strategies of all players is proposed. By leveraging the neural network approximation and gradient descent method, simultaneous continuous‐time adaptive tuning laws are provided for both critic and actor neural network weights. These laws facilitate the estimation of the optimal value function and optimal policy without requiring knowledge or identification of the system's dynamics. The closed‐loop system stability and convergence of weights are guaranteed through the Lyapunov analysis. Additionally, the algorithm is enhanced to reduce the number of auxiliary NNs used in the critic. The simulation results for a two‐player non‐zero‐sum game validate the effectiveness of the proposed algorithm.

Measures for the Importance of Information Exchange in Linear Quadratic Differential Games Over Networks

The objective of this paper is to study and characterize the role and the importance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information</i> in achieving a feedback (Nash) equilibrium strategy in linear quadratic (LQ) differential games whenever the underlying players are distributed over a (physical or logic) network. It is assumed that each player should achieve a desired goal, quantified by an individual cost functional, in a competitive dynamic environment - captured by an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> - by relying only on the information and data exchanged with other players according to a prescribed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network:</i> the objective of the paper is to establish the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">value</i> of such an information exchange pattern towards achieving a more favorable (social) equilibrium. Interestingly, it is not assumed that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network</i> coincide. Moreover, since the ability of achieving a certain Nash equilibrium strategy may be lost even by removing a single communication link in the network - thus partially limiting the use of the metrics discussed above - in the second part of the paper we also consider the value of the information in forming approximate Nash equilibrium strategies, namely the so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -Nash equilibria. Finally, the newly defined metrics are corroborated - together with a few suggested constructive results to characterize such values - by means of numerical examples.

3-Strategy evolutionary game model for operation extensions of subway networks

With the growing demand for night travel and the development of the night-time economy, extending the operation time of metropolitan subway networks has become essential. This paper proposes a 2-population 3-strategy evolutionary game model for subway network operation extension decisions, which considers the interests of subway companies, passengers, and the government while addressing transfer failure between lines due to uncoordinated subway operation time adjustments. The study resolves the Evolutionary Equilibria (EEs) of the replicator dynamics system and identifies key factors influencing the evolutionary path using theoretical calculations and a ternary phase diagram. Additionally, an extended game model investigates the impact of adjustment minutes. To verify the models, a case study based on the Beijing subway network is conducted. We show that the rule of maximum total fitness ensures the priority of EE strategies for all lines, selecting Nash equilibrium strategies for the lines which cannot select EE in a vastly reduced network strategy profile space (from 729 to 2). The study concludes that combining subsidy measures can eliminate multiple equilibria produced by implementing only penalty measures, and combining penalty measures can have the effect of making small subsidies as effective as large subsidies. The variation of the satisfaction/loyalty affects the system equilibrium only when both game players value passenger satisfaction/loyalty. Finally, we find that the proper adjustment minutes not only can make operation extension a long-term best strategy but also increase the payoff (106,158 RMB/year) and enhance the occupancy rate (20 %) of the last train. The study provides a theoretical foundation for the government to regulate the operation extension of subway companies from multiple perspectives.

Relationship between Nash Equilibrium Strategies and H2/H∞ Control of Mean-Field Stochastic Differential Equations with Multiplicative Noise

The relationship between finite-horizon mean-field stochastic H2/H∞ control and Nash equilibrium strategies is investigated in this technical note. First, the finite-horizon mean-field stochastic bounded real lemma (SBRL) is established, which is key to developing the H∞ theory. Second, for mean-field stochastic differential equations (MF-SDEs) with control- and state-dependent noises, it is revealed that the existence of Nash equilibrium strategies is equivalent to the solvability of generalized differential Riccati equations (GDREs). Furthermore, the existence of Nash equilibrium strategies is equivalent to the solvability of H2/H∞ control for MF-SDEs with control- and state-dependent noises. However, for mean-field stochastic systems with disturbance-dependent noises, these two problems are not equivalent. Finally, a sufficient and necessary condition is presented via coupled matrix-valued equations for the finite-horizon H2/H∞ control of mean-field stochastic differential equations with disturbance-dependent noises.

Plants’ competition under autotoxicity effect: an evolutionary game

We develop a 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} evolutionary matrix game to model vegetation dynamics due to the effect of autotoxicity. The phenomenon of autotoxicity refers to the rise in soil of negative conditions for plant performance induced by the plants themselves. Relating the Nash Equilibrium Strategies of the game to the stability of the equilibrium points of the induced population dynamics, we investigate under which conditions coexistence of low and highly sensitive to autotoxicity plants occurs and under which a monospecific population dominates the competition. Based on this classification, we investigate the optimal distribution of the two distinct types of plants in order to maximize the cumulative total fitness and determine if this distribution is stable. The primary outcome of this study is to analyze the necessary conditions for achieving the highest total fitness in both mixed and monospecific populations of low-sensitivity plants. In contrast, we argue that a monospecific population of highly sensitive plants can never maximize overall fitness.

Multi-objective scheduling in dynamic of household paper workshop considering energy consumption in production process

Aim: The uncertainty and complexity of the production process of household paper are growing sharply in modern factories. Due to the influence of rising energy costs and environmental policies, the demand for reducing production costs and energy consumption is also increasing. Therefore, it is studied that the dynamic shop scheduling problem of household paper production considering simultaneously the cost with energy consumption. Methods: A mathematical model of the multi-objective and multi-constraint household paper scheduling problem is established first. The multi-objective scheduling process is transformed into a multi-agent Markov game process by assigning each objective to each agent. Upon which, a multi-agent game model is constructed for the household paper scheduling problem based on deep reinforcement learning; it is a proposed D3QN algorithm and Nash equilibrium strategy, and the state characteristics and action selection space are proposed according to the characteristics of the household paper production. The model performance has been verified by the actual production data. Results: Results show that the proposed method not only achieves better performance than traditional scheduling methods but also persists in its advantages even when the configuration of the manufacturing system changes. Conclusion: Multi-agent deep reinforcement learning in the Markov game has a good prospect in solving multi-objective dynamic scheduling problems for household paper production.

A Game-Theoretical Approach to Clinical Decision Making with Immersive Visualisation

Cancer is a disease characterised by changes in combinations of genes within affected tumour cells. The deep understanding of genetic activity afforded to cancer specialists through complex genomics data analytics has advanced the clinical management of cancer by using deep machine learning algorithms and visualisation. However, most of the existing works do not integrate intelligent decision-making aids that can guide users in the analysis and exploration processes. This paper contributes a novel strategy that applies game theory within a VR-enabled immersive visualisation system designed as the decision support engine to mimic real-world interactions between stakeholders within complex relationships, in this case cancer clinicians. Our focus is to apply game theory to assist doctors in the decision-making process regarding the treatment options for rare-cancer patients. Nash Equilibrium and Social Optimality strategy profiles were used to facilitate complex analysis within the visualisation by inspecting which combination of genes and dimensionality reduction methods yields the best survival rate and by investigating the treatment protocol to form new hypotheses. Using a case simulation, we demonstrate the effectiveness of game theory in guiding the analyst with a patient cohort data interrogation system as compared to an analyst without a decision support system. Particularly, the strategy profile (t-SNE method and DNMT3B_ZBTB46_LAPTM4B gene) gains the highest payoff for the two doctors.