Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrixcan be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.

Design and pricing of maintenance service contract based on Nash non-cooperative game approach

Nowadays manufacturers are facing fierce challenge.Apart from the products,providing customers with multiple maintenance options in the service contract becomes more popular,since it can help to improve customer satisfaction,and ultimately promote sales and maximize profit for the manufacturer.By considering the combinations of corrective maintenance and preventive maintenance,totally three types of maintenance service contracts are designed.Moreover,attractive incentive and penalty mechanisms are adopted in the contracts.On this basis,Nash non-cooperative game is applied to analyze the revenue for both the manufacturer and customers,and so as to optimize the pricing mechanism of maintenance service contract and achieve a win-win situation.Numerical experiments are conducted.The results show that by taking into account the incentive and penalty mechanisms,the revenue can be improved for both the customers and manufacturer.Moreover,with the increase of repair rate and improvement factor in the preventive maintenance,the revenue will increase gradually for both the parties.

Nash Equilibrium of a Fixed-Sum Two-Player Game

It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.

基于随机-Nash-Harsanyi讨价还价博弈的光伏-氢储能多主体分散协同调度

将光伏(PV)主体与氢储能(HES)主体结合,形成PV-HES联合系统,是促进可再生能源就地消纳,实现能源低碳转型的有效措施。针对光伏出力随机干扰及主体互异等问题,提出PV-HES的分散协同调度模型。首先,基于Nash-Harsanyi讨价还价理论建立PV-HES的多主体合作运行模型,并将其等效转换为电能交易量确定子问题和电能交易成本确定子问题;然后,为了保护各主体隐私,运用改进的交替方向乘子法对上述两个子问题进行分布式求解;最后,在一个典型的PV-HES联合系统中进行算例仿真,以验证所提分散协同调度模型以及分布式求解算法的有效性。仿真结果表明,氢储能主体的运营成本分别降低了11.67%和10.48%,光伏运营商的收益提高了15.72%。此外,通过对交替方向乘子法的改进也提高了系统的求解效率。

不连续多目标博弈弱Pareto-Nash均衡解的存在性与Hadamard良定性

该文主要考虑不连续多目标博弈问题解的存在性和良定性.首先将不连续条件更优回应保障推广到n人多目标博弈,证明了其弱Pareto-Nash均衡点的存在性定理.更进一步地,研究了此类不连续n人多目标博弈的Hadamard良定性.这些结果推广了P.J.Reny等的研究成果.

Infinite horizon H-two/H-infinity control for descriptor systems:Nash game approach

This paper studies the infinite time horizon mixed H-two/H-infinity control problem for descriptor systems using Nash game approach. A necessary/sufficient condition for the existence of infinite horizon H-two/H-infinity control is presented in the form of two coupled algebraic Riccati equations, respectively. Finally, a suboptimal H-two/H-infinity controller design is given based on an iterative linear matrix inequality algorithm.

Control and Nash Games with Mean Field Effect

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007,of Gueant-Lasry-Lions in 2011,of HuangCaines-Malham in 2007 and many others).There are a lot of applications.In general,the applications concern approximating an infinite number of players with common behavior by a representative agent.This agent has to solve a control problem perturbed by a field equation,representing in some way the behavior of the average infinite number of agents.This approach does not lead easily to the problems of Nash equilibrium for a finite number of players,perturbed by field equations,unless one considers averaging within different groups,which has not been done in the literature,and seems quite challenging.In this paper,the authors approach similar problems with a different motivation which makes sense for control and also for differential games.Thus the systems of nonlinear partial differential equations with mean field terms,which have not been addressed in the literature so far,are considered here.

Electromagnetic buffer optimization based on Nash game

Aiming at the problem of unstable buffering process of electromagnetic buffer(EMB)under intensive impact load,a three-segment electromagnetic buffer is proposed.The inner tube and air-gap of EMB are divided into three segments.The finite element analysis and impact test results show that the resultant resistance force(RRF)curve has two hump-shaped peaks,which is the reason for the unstable buffering process.In order to stabilize the buffering process,a multi-objective optimization design method of EMB based on Nash game theory is proposed.Firstly,the optimization model is established by taking the two peaks of the RRF curve and the maximum buffer displacement as the optimization objectives.Secondly,the multi-objective optimization model is transformed into a game model by sensitivity analysis and fuzzy clustering.Then,a Nash equilibrium solution strategy of EMB Nash game model based on symmetric elitist information exchange is proposed,which integrates gene expression programming(GEP)surrogate model and genetic algorithm(GA)as an optimization solver.Finally,the Nash equilibrium of the game model is obtained.The results show that the smoothness of the RRF curve has been significantly improved,which proves the effectiveness of the game strategy.

A Nash Game Approach to Mixed H_2/H_∞ Model Predictive Control: Part 3–Output Feedback Case

In this paper, the state-feedback Nash game based mixed H2/H∞ design^（[1, 2]）has been extended for output feedback case. The algorithm is applied to control bioreactor system with a Laguerre-Wavelet Network（LWN）^（[3, 4]）model of the bioreactor.This is achieved by using the LWN model as a deviation model and by successively linearising the deviation model along the state trajectory. For reducing the approximation error and to improve the controller performance, symbolic derivation algorithm, viz.,automatic differentiation is employed. A cautionary note is also given on the fragility of the output feedback mixed H2/H∞ model predictive controller^（[4, 5]）due to its sensitivity to its own parametric changes.

On the finite horizon Nash equilibrium solution in the differential game approach to formation control

The solvability of the coupled Riccati differential equations appearing in the differential game approach to the formation control problem is vital to the finite horizon Nash equilibrium solution.These equations(if solvable)can be solved numerically by using the terminal value and the backward iteration.To investigate the solvability and solution of these equations the formation control problem as the differential game is replaced by a discrete-time dynamic game.The main contributions of this paper are as follows.First,the existence of Nash equilibrium controls for the discretetime formation control problem is shown.Second,a backward iteration approximate solution to the coupled Riccati differential equations in the continuous-time differential game is developed.An illustrative example is given to justify the models and solution.

基于改进Nash-Q均衡迁移算法的源网荷储协同优化策略

为了充分发挥多类型储能资源的调度潜力,实现源网荷储协同优化调度,提出了计及电池储能、抽水蓄能、电动汽车的多类型储能调度策略。以低碳经济为目标,构建了考虑多主体博弈的源网荷储协同优化调度模型。为了在保证源网荷三侧整体利益的同时兼顾自身利益,基于Nash均衡理论,利用强化迁移学习技术,提出了一种基于改进Nash-Q的均衡迁移算法。利用K-means聚类使数据离散化,增设双结构经验池以提高样本利用率,从而有效提高了模型的泛化能力。基于实际区域电网的数据进行仿真验证,结果表明所提策略能有效降低系统的经济成本和碳处理费用,提高新能源消纳能力。

Fuzzy Nash equilibrium of fuzzy n-person non-cooperative game

The fuzzy non-cooperative game with fuzzy payoff function is studied. Based on fuzzy set theory with game theory, the fuzzy Nash equilibrium of fuzzy non-cooperative games is proposed. Most of researchers rank fuzzy number by its center of gravity or by the real number with its maximal membership. By reducing fuzzy number into a real number, we lose much fuzzy information that should be kept during the operations between fuzzy numbers. The fuzzy quantities or alternatives are ordered directly by Yuan＇s binary fuzzy ordering relation. In doing so, the existence of fuzzy Nash equilibrium for fuzzy non-cooperative games is shown based on the utility function and the crisp Nash theorem. Finally, an illustrative example in traffic flow patterns of equilibrium is given in order to show the detailed calculation process of fuzzy Nash equilibrium.

Polynomial Time Method for Solving Nash Equilibria of Zero-Sum Games

There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.

考虑多重不确定性与电碳耦合交易的多微网合作博弈优化调度

双碳背景下,构建低碳运行的能源系统是实现“双碳”目标的重要方向与实施路径。为了促进多微网系统内部能源的本地消纳以及低碳经济运行,该文对不确定性环境下多微网系统的合作运行及电碳耦合交易展开研究。首先,对于每个微网,构建了电转气碳捕集系统相耦合的热电联产运行模式,基于地方碳交易市场和阶梯碳交易机制,提出了多能源微网的低碳运行模型;其次,考虑到新能源发电和电力市场电价都存在不确定性的实际情况,采用机会约束和鲁棒优化的方法,以降低不确定性影响;再次,基于纳什谈判理论,建立了多个微网电碳耦合的合作博弈模型,各微网主体可同时参与到上级能源市场和地方能源市场中进行电能和碳排放配额的交易;最后,将非凸的合作博弈问题分解为两个线性可求解的子问题,进一步采用交替方向乘子法对问题进行求解。通过算例验证,该文所提方法可以有效提升各微网经济效益并减少碳排放。

有限理性与一类群体博弈弱有效Nash均衡的稳定性

该文引入了一个参与人具有无限个目标的群体博弈模型,并且定义了它的弱有效Nash均衡的概念,进一步又证明了该弱有效Nash均衡的存在性,最后构造了一个有限理性模型研究了有限理性下该弱有效Nash均衡的稳定性.

Statistical Learning in Game Theory

In economics, buyers and sellers are usually the main sides in a market. Game theory can perfectly model decisions behind each “player” and calculate an outcome that benefits both sides. However, the use of game theory is not lim-ited to economics. In this paper, I will introduce the mathematical model of general sum game, solutions and theorems surrounding game theory, and its real life applications in many different scenarios.

基于Nash博弈的分布式驱动电动汽车横摆稳定性协调控制策略

为提高轮毂电机驱动电动汽车在高速、低附着等危险工况下的侧向稳定性,提出一种基于Nash博弈的协同控制策略,采用上下双层控制结构进行稳定性控制策略的设计。上层引入Nash博弈协调控制策略决策前轮转角和附加横摆力矩,跟踪期望横摆角速度和质心侧偏角;下层根据轴荷比例分配四个车轮的驱动力矩。并在CarSim/Simulink的联合仿真平台进行危险工况下双移线仿真试验,结果表明,相比于只进行主动前轮转向控制,在潮湿沥青路面以75 km/h行驶时,采用基于Nash博弈的协调控制策略横摆角速度最大误差为2.25°,侧向速度最大误差为0.12 m/s,且保持良好的路径跟踪性能;通过适当协调主动前轮转向(AFS)和直接横摆力矩控制(DYC)的动作,文章所提出的控制策略可以有效地提高横向稳定性,保证车辆在危险行驶工况下正常行驶。

基于三层博弈的社区综合能源系统电热交易策略

随着综合能源系统逐渐向智能化、清洁化、去中心化转型,发用电一体的产消者应运而生。为实现综合能源系统可再生能源的充分消纳,提高各主体的交互效益,提出基于三层博弈的社区综合能源系统电热交易策略。首先,建立基于社区合作博弈嵌套的综合能源运营商与能源社区主从博弈模型,运营商以最大化自身效益为目标制定分时电价、热价并传递给能源社区。能源社区响应运营商的电热决策,以消费剩余最大为目标制定与其他社区的交互策略,确定购能需求并传递给运营商。其次,为降低综合能源运营商碳排放,充分消纳各运营商的资源剩余,建立多综合能源运营商的纳什谈判模型。然后,利用Karush-Kuhn-Tucker(KKT)条件将模型进行转化,并利用McCormick法进行凸松弛。最后,采用交替方向乘子分布式算法对模型进行求解,通过算例验证了所提策略能够有效提升各主体交互效率、降低综合能源系统碳排放、提高各主体效益。

基于纳什谈判和主从博弈的多园区综合能源系统优化调度

为充分考虑多园区综合能源系统(MCIES)中多利益主体间交易合作与多种能量间的耦合关系,提出了1种基于纳什谈判-主从博弈的点对点分布式交易策略。首先,建立考虑综合需求响应的MCIES主从博弈动态定价模型,以园区综合能源系统(CIES)运营商作为领导者,给下层用户制定动态电价与热价,用户响应电价与热价进行综合需求响应,CIES运营商根据用户的需求响应计划调整机组出力;其次,基于纳什谈判理论,建立包含主从博弈动态定价机制与CIES间合作博弈的MCIES双层优化模型;然后,将原问题等效为MCIES收益最大化子问题和合作收益分配最大化子问题,采用交替方向乘子法(ADMM)对模型进行求解。最后,仿真结果表明所提策略可实现CIES联盟利润的最大化,并合理分配收益,同时最小化用户用能成本。

考虑碳排放分摊的综合能源服务商交易策略

为寻求在满足用户用能需求的同时降低系统碳排放的有效途径,提出了一种考虑碳排放分摊的多综合能源服务商能源交易策略。基于网络流模型实现碳流追踪并联合Shapley理论建立了综合能源服务商碳排放成本计算模型。应用Nash议价理论构建了多综合能源服务商合作博弈运行模型,将综合能源服务商碳排放成本引入优化目标函数,并利用分布式算法求解最优能源交易量和交易费用。基于由修改的IEEE 39节点电网和12节点气网构成的多综合能源服务商系统进行仿真分析,验证了所提模型和方法的有效性。