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.

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.

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.

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。

Game Theory Model and Equilibrium Analysis of Peasant's Production Decision

Unbalanced agricultural production decision becomes the great block that influences the effective distribution of social resources, national grain security, social stability and economic development. This paper took the game theory as an analyzed tool to describe the interactional processes among the peasants, and set up the game theory model of independent decision and joint decision by peasants. It was shown that the government＇s positive guide and the market environment macroscopically controlled by the government could effectively increased the peasants＇ income

QUASI-EQUILIBRIUM PROBLEMS AND CONSTRAINED MULTIOBJECTIVE GAMES IN GENERALIZED CONVEX SPACE

A class of quasi-equilibrium problems and a class of constrained multiobjective games were introduced and studied in generalized convex spaces without linear structure. First, two existence theorems of solutions for quasi-equilibrium problems are proved in noncompact generalized convex spaces. Then, ar applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact generalized convex spaces. These theorems improve, unify, and generalize the corresponding results of the multiobjective games in recent literatures.

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.

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.

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.

基于博弈论的弹目攻防决策方法研究

针对空战环境中弹目攻防双方的对抗特性,提出了一种基于博弈论的弹目攻防决策方法。基于导弹目标运动数学关系得到状态方程,根据弹目攻防对抗机理建立“一对一导弹-目标”动态博弈模型,确定弹目双方策略集与收益矩阵,提出混合策略纳什均衡求解方法,并结合模型滚动预测方法获得该策略空间的纳什均衡点。算例仿真结果表明,基于混合策略下导弹制导律为该策略空间的纳什均衡点,且该方法可以减小导弹对目标的脱靶量,提高导弹的命中精度,为导弹攻防作战提供了依据。

On Topics in Quantum Games

This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.

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。

航天器追踪-逃逸-防御三方博弈均衡解

为提高航天器在轨防御能力,对航天器追踪-逃逸-防御三方博弈场景进行了研究。博弈中,追踪航天器(追踪器)旨在拦截逃逸航天器(逃逸器),逃逸器旨在躲避追踪器,而防御航天器(防御器)通过主动拦截追踪器来保护逃逸器。由于防御器的存在,追踪器在接近目标时不得不规避防御器,而逃逸器和防御器之间也可以开展合作。对于这样的场景,综合能量和距离指标,建立了线性二次型微分对策模型,推导了三方博弈的纳什均衡条件,求解了追踪器的最优追踪制导律和逃逸器与防御器的最优合作逃逸-防御制导律。进一步考虑多防御器的场景,求解了三方多人博弈的纳什均衡解。仿真结果表明,防御器的存在提高了逃逸器的生存能力,二者可以在机动加速度劣势情况下共同抵御一个机动能力强的追踪器。处在绕飞护卫下的防御器的初始位置并非离追踪器或逃逸器越近越好,而是存在优势位置。

基于免疫粒子群算法的非合作博弈Nash均衡问题求解

针对N人非合作博弈Nash均衡求解问题,将免疫算法中抗体浓度抑制机制和免疫记忆功能引入基本粒子群算法,提出了一种求解博弈问题Nash均衡的免疫粒子群算法。该算法通过抗体浓度抑制机制和免疫记忆功能来保持种群的多样性,不仅保持了粒子群算法简单、易于实现的特点,而且增强了粒子群算法的全局寻优能力,加快了算法的速度。实验表明,提出的算法具有较好的性能,优于免疫算法和基本粒子群算法。

一种求解NASH均衡解的遗传算法

将现代生物遗传基因理论引入到对策问题的均衡解计算中来 ,本文设计了求解有限n人非合作对策的 Nash均衡解的遗传算法模型 ,并通过双矩阵对策的例题验证了算法的有效性。

航天器姿态控制群体博弈分布式分配方法

针对由多个细胞星构成的组合卫星的姿态控制问题,提出了一种包含控制力矩计算层与冗余力矩分配层的双层姿态控制方法。将细胞星间的冗余力矩分配问题转化为群体博弈的策略选择问题,综合考虑能量消耗、飞轮输出力矩裕度和飞轮角动量裕度设计收益函数,以力矩分配系数作为群体质量,以细胞星作为策略建立群体博弈模型。考虑群体演化中模仿和比较形成的不同演化策略,分别采用Smith动力学以及复制器动力学修订协议求解博弈的Nash均衡解,并利用有限制策略的群体博弈理论形成分布式冗余力矩分配方式,得到各个细胞星的控制策略。最后通过仿真实验验证了此分配方式的有效性。

无人机-无人车跨域编队跟踪的分布式博弈策略

针对无人机-无人车跨域系统的编队跟踪与扰动抑制问题,开展了领-从架构下考虑存在扰动的跨域编队跟踪的分布式博弈策略研究。首先,建立了含扰动的无人机-无人车编队跟踪模型,每架无人机只使用局部信息来最小化自己的性能指标,而整个跨域系统试图寻求基于通信拓扑的最优决策。其次,提出了一种最差扰动下的博弈策略,并严格证明了系统中存在纳什均衡。然后,引入了一个分布式终端状态观测器,得到了所提出博弈策略的局部实现。最后,搭建了一套无人机-无人车跨域系统验证平台,并通过一个跨域编队跟踪应用案例验证了所提出分布式博弈策略不仅可以有效地抑制扰动,还能形成期望编队队形,并且性能指标在数值仿真和实物试验下的差异不超过0.6%,证明所提出分布式博弈策略有良好的实用性。

激励发电节能减排交易的Nash均衡两级优化模型

电力交易中心需要设计一种机制,将发电企业的环境成本内部化,从而在企业追求经济效益的过程中实现社会环境效益最大化。建立了燃煤发电企业节能减排交易两级优化模型,第一级优化模型将社会环境效益和发电节能效益最大化作为目标函数,并考虑机组出力约束、负荷约束和排污量约束等条件;第二级优化模型以各个发电企业的竞争利益最大化作为目标函数,以第一级模型的KKT条件作为约束。该模型反应出的边际排污激励价格机制,既可以从总量上控制SO2的排放、节约能耗,又提高了燃煤发电企业进行排污治理的积极性。算例分析表明,模型和算法可以实现发电节能减排、激励竞争的效果。

基于Nash均衡的电力市场竞价策略分析

描述了在双边电力市场中基于Nash均衡的竞价策略 ,这种策略的前提条件是使整个交易过程中的总成本最小化 ,问题的关键是找到最佳的买方和卖方的组合 ,即Nash均衡 .介绍了电力市场的运营模式 ,阐述了竞价策略的数学模型以及模型的参数估计和基本假设 ,算例以及结论进一步说明了这种竞价策略 ,并证明了策略的Nash均衡的存在 .