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.

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。

Analysis of Nash-Cournot equilibrium for electricity markets considering option contracts

Option contract is one of the most important instruments for power generators developing bidding strategies and hedging market risk. Based on the peculiarities of bid-based-pool （BBP） power markets, a joint two-stage Cournot equilibrium model for option and spot markets is developed, and analytical formulas for market equilibrium are derived using a backward induction method. The impacts of option contract on efficiency of electricity markets and the behaviors of strategic generators are analyzed. The results show that strategic generators will voluntarily participate in strategic option contracting, and the existence of option contract accelerates the degree of competitive intensity in electricity markets and mitigates the market power abuse of generators to a large extent. In order to retain high spot market price and stable revenues, generators are interested in holding extremely high volatility of spot market price.

A Statistical Analysis of Games with No Certain Nash Equilibrium Make Many Results Doubtful

Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.

Generalized Quantum Games with Nash Equilibrium

We define generalized quantum games by introducing the coherent payoff operators and propose a simple scheme to illustrate it.The scheme is implemented with a single spin qubit system and a two-entangled-qubit system.The Nash Equilibrium Theorem is proved for the models.

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand

Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.

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。

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.

广义伪连续下的多主从博弈弱Pareto-Nash均衡的稳定性研究

在现实生活中,许多决策问题都包含多个参与者和多个目标,如企业间的合作博弈、资源共享博弈等。通过探究广义多目标多主从博弈的稳定性,可以制定出更加有效的决策策略和机制,确保参与者能够长期稳定地获得收益并维持合作关系。文章针对Hausdorff拓扑空间内的一种特定的广义多主体-多从体博弈,当其中的参与者支付函数满足广义伪连续性时,研究证明广义多主体-多从体博弈的弱Pareto-Nash均衡解映射存在上半连续性。同时,当参与者支付函数在广义上呈现上伪连续时,也证实了广义多主体-多从体博弈的弱Pareto-Nash均衡解映射的上半连续性。应用本质解的定义,可揭示广义多主体-多从体博弈下弱Pareto-Nash均衡解的稳定性。