Distributed Nash Equilibrium Seeking Strategies Under Quantized Communication

This paper is concerned with distributed Nash equi librium seeking strategies under quantized communication. In the proposed seeking strategy, a projection operator is synthesized with a gradient search method to achieve the optimization o players' objective functions while restricting their actions within required non-empty, convex and compact domains. In addition, a leader-following consensus protocol, in which quantized informa tion flows are utilized, is employed for information sharing among players. More specifically, logarithmic quantizers and uniform quantizers are investigated under both undirected and connected communication graphs and strongly connected digraphs, respec tively. Through Lyapunov stability analysis, it is shown that play ers' actions can be steered to a neighborhood of the Nash equilib rium with logarithmic and uniform quantizers, and the quanti fied convergence error depends on the parameter of the quan tizer for both undirected and directed cases. A numerical exam ple is given to verify the theoretical results.

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.

具有加性耦合效用和连续统参与人博弈中的强Nash均衡

本文将研究具有加性耦合效用和连续统参与人博弈中的强Nash均衡。我们首先证明具有加性耦合效用和有限参与人博弈中强Nash均衡的存在性。进一步,对具有加性耦合效用和连续统参与人的博弈,我们引入弱强Nash均衡的概念,并证明它的存在性定理。本文发展了强Nash均衡的研究。

转移概率部分未知的离散时间Markov跳变系统Nash微分博弈

考虑到转移概率矩阵元素无法完全获悉,如何在转移概率部分未知的情境下研究离散时间Markov跳变系统Nash微分博弈是有待解决的问题之一,这一问题可以为转移概率部分未知的Markov跳变系统Nash微分博弈理论在管理问题上的应用提供理论支撑。基于此,本文首先研究单人博弈情形,即ε-次优控制问题,借助自由连接权矩阵和配方法,得到了ε-次优控制策略存在的充分性条件,并给出了成本函数上界的显式表达;然后延伸至双人博弈进行分析,得到了ε-次优Nash均衡策略存在的条件等价于求解双线性矩阵不等式和矩阵不等式的优化问题,并通过启发式算法求解优化问题得到ε-次优Nash均衡策略;最后通过数值算例证明了主要结论的有效性。

Distributed Nash Equilibrium Seeking for General Networked Games With Bounded Disturbances

This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information.First,reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with firstorder and second-order players,respectively.In the developed algorithms,the observed disturbance values are included in control signals to eliminate the influence of disturbances,based on which a gradient-like optimization method is implemented for each player.Second,a signum function based distributed algorithm is proposed to attenuate disturbances for games with secondorder integrator-type players.To be more specific,a signum function is involved in the proposed seeking strategy to dominate disturbances,based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players'objective functions.Through Lyapunov stability analysis,it is proven that the players'actions can approach a small region around the Nash equilibrium by utilizing disturbance observerbased strategies with appropriate control gains.Moreover,exponential(asymptotic)convergence can be achieved when the signum function based control strategy(with an adaptive control gain)is employed.The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform(V-REP)and MATLAB.

Fully Distributed Nash Equilibrium Seeking for High-Order Players With Actuator Limitations

This paper explores the problem of distributed Nash equilibrium seeking in games, where players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: (1) ensuring the boundedness of control inputs;(2) avoiding any global information/parameters;and (3) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.

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.

Heston模型下的两人鲁棒非零和随机微分投资组合博弈

用Heston模型描述风险资产的价格动态,构建了包含一种无风险资产和一种风险资产的金融市场,投资者可以将其财富自由地配置于无风险资产和风险资产中.考虑到投资者之间经济行为的随机博弈,用相对业绩刻画投资者之间的博弈行为,同时考虑模型的不确定性,以最大化最坏情境下投资者相对业绩的期望效用为目标,构建了包含两个投资者的鲁棒非零和随机微分投资组合博弈模型,利用动态规划方法分别求得了CRRA效用下Nash均衡策略的解析表达,借助数值仿真算例进行了参数的敏感性分析并给出了相应的经济意义阐释.研究发现:相较于不涉及市场竞争的传统投资策略,竞争将使投资者产生羊群效应,跟风投资风险资产,致使金融市场的系统性风险上升.此外,与不考虑模型不确定性相比,模型的不确定性使得投资者减少对风险资产的投资.

SCR脱硝系统的RBF条件积分滑模参数优化控制

针对选择性催化还原(SCR)脱硝系统在火电机组深度调峰下因系统特性随着机组负荷改变而难以控制的问题,提出基于径向基函数(RBF)的条件积分滑模优化控制方案,用条件积分滑模减少抖振与调节时间,用RBF实时逼近SCR脱硝系统特性改变下的未知扰动以提高系统鲁棒性。并提出一种纳什均衡量子粒子群寻优(NEQPSO)算法,以获得RBF条件积分滑模控制方案的最优参数,并在20%~100%的机组负荷下进行仿真实验。结果表明:优化后的RBF条件积分滑模控制与传统控制方案相比超调量降低50.26%、调节时间缩短13.55%,在干扰信号下超调量仅有1.50%,其响应速度更快、抗干扰能力更强、鲁棒性更好。

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。

不完全信息Epsilon纳什均衡的航天器末端追逃博弈策略

针对不完全信息下的航天器末端追逃问题,提出了一种满足Epsilon纳什均衡的微分博弈控制策略。首先,建立了完全信息下的有限时间追逃纳什均衡策略对,并将其作为目标航天器实际采取的控制策略,使目标航天器掌握博弈进程的完全信息,进而获得更好的逃逸性能。在此基础上,考虑拦截航天器不能获取目标控制矩阵信息的态势,设计了基于广义卡尔曼滤波的行为学习信息估计算法,使拦截器能够对目标的不完全信息进行估计,并提出了不完全信息下的末端追逃博弈控制策略。经过理论分析,证明了所设计的不完全信息下微分博弈策略对满足Epsilon纳什均衡。最后,仿真结果表明该算法可以有效估计目标的不完全信息,确保拦截器能够在有限时间内快速拦截目标。

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.

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.

A Penalty Approach for Generalized Nash Equilibrium Problem

The generalized Nash equilibrium problem （GNEP） is a generalization of the standard Nash equilibrium problem （NEP）, in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players. This problem has been used to model various problems in applications. However, the convergent solution algorithms are extremely scare in the literature. In this paper, we present an incremental penalty method for the GNEP, and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs. We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.

An Extension of One-Period Nash Equilibrium Model in Non-Life Insurance Markets

This paper deals with an extension of the one-period model in non-life insurance markets (cf. [1]) by using a transition probability matrix depending on some economic factors. We introduce a multi-period model and in each period the solvency constraints will be updated. Moreover, the model has the inactive state including some uninsured population. Similar results on the existence of premium equilibrium and sensitivity analysis for this model are presented and illustrated by numerical results.

On Dynamics and Nash Equilibriums of Networked Games

Networked noncooperative games are investigated,where each player(or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed.The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.

基于序贯Nash谈判博弈的石油开采策略研究

考虑国外环保组织与石油开采公司游说发展中国家政府制定石油开采政策的情形,构建序贯非对称Nash谈判博弈模型:政府首先与国外环保组织以及石油开采公司进行三方谈判;如果三方谈判破裂,则政府选择一方进行双边谈判,谈判破裂点为政府与另一方进行双边谈判的结果;如果后者的双边谈判破裂,则政府选择最大化国家福利的石油开采政策。研究表明:在三方谈判中,如果一个游说团体筹集资金的成本低或政府对其缴纳的费用给予更高的评估,则该游说团体对政府决策具有更大的影响;三方谈判破裂后对政府决策仍产生影响的游说团体在三方谈判均衡中所支付的费用低于在双边谈判均衡中所支付的费用;当环保组织与政府之间不存在利益冲突时,环保组织可能受益于其维护三方谈判的行为。

基于改进TOPSIS算法的低压多台区柔性互联运行评价方法研究

为解决光伏高渗透低压台区的电压越限、三相不平衡等问题,引入含储能柔性互联装置(FID),实现台区间的功率互济与优化运行。针对低压多台区柔性互联的运行评价方法,提出一种基于AHP-CRITIC组合赋权的改进TOPSIS评价模型。首先,从含储能FID接入对台区改善较为明显的运行电压、运行功率、运行经济3个方面建立评价体系;其次,通过AHP法与CRITIC法确定主、客观权重,利用纳什均衡理论实现AHP-CRITIC组合赋权;然后,基于加权灰色关联度改进TOPSIS,进一步反映指标内部变化。最后,以实际台区为算例,分析所提方法的有效性。

Distributed Optimal Variational GNE Seeking in Merely Monotone Games

In this paper, the optimal variational generalized Nash equilibrium(v-GNE) seeking problem in merely monotone games with linearly coupled cost functions is investigated, in which the feasible strategy domain of each agent is coupled through an affine constraint. A distributed algorithm based on the hybrid steepest descent method is first proposed to seek the optimal v-GNE. Then, an accelerated algorithm with relaxation is proposed and analyzed, which has the potential to further improve the convergence speed to the optimal v-GNE. Some sufficient conditions in both algorithms are obtained to ensure the global convergence towards the optimal v-GNE. To illustrate the performance of the algorithms, numerical simulation is conducted based on a networked Nash-Cournot game with bounded market capacities.

Optimal confrontation position selecting games model and its application to one-on-one air combat

In the air combat process,confrontation position is the critical factor to determine the confrontation situation,attack effect and escape probability of UAVs.Therefore,selecting the optimal confrontation position becomes the primary goal of maneuver decision-making.By taking the position as the UAV’s maneuver strategy,this paper constructs the optimal confrontation position selecting games(OCPSGs)model.In the OCPSGs model,the payoff function of each UAV is defined by the difference between the comprehensive advantages of both sides,and the strategy space of each UAV at every step is defined by its accessible space determined by the maneuverability.Then we design the limit approximation of mixed strategy Nash equilibrium(LAMSNQ)algorithm,which provides a method to determine the optimal probability distribution of positions in the strategy space.In the simulation phase,we assume the motions on three directions are independent and the strategy space is a cuboid to simplify the model.Several simulations are performed to verify the feasibility,effectiveness and stability of the algorithm.