基于半张量积方法的网络化博弈的分析、控制及应用

发布时间：2018-01-30 20:53

本文关键词： 网络演化博弈纳什均衡策略优化协同演化传染病动态矩阵半张量积　出处：《山东大学》2016年博士论文　论文类型：学位论文

【摘要】：近年来随着复杂网络的快速发展,网络化博弈已经成为博弈论研究中的一个热门课题.与传统博弈相比,网络化博弈用网络拓扑中的顶点表示玩家,用边表示相邻玩家间存在博弈关系,每个玩家仅与周围的邻居进行博弈并获得累积收益,这与社会实际更加相符.目前在生物学、经济学、国际关系、计算机科学、军事战略和其他诸多领域都得到了广泛的应用,有着极其重要的理论价值和研究意义.在有限网络演化博弈中,每个玩家有有限个策略可以选择,其动态演化过程与逻辑网络动态有着天然的联系.而矩阵半张量积作为处理有限集上动态的有力工具,也可以用来研究有限网络演化博弈问题.利用该方法可以将博弈动态转化为代数形式,为其建立严格的数学框架,这有利于人们对实际生活中的博弈现象产生更精确更深入的认识,进而指导人们对博弈进行预测和控制.鉴于网络演化博弈问题的多样性,半张量积理论在网络演化博弈领域还有广阔的应用空间,值得进一步研究.本文利用矩阵半张量积方法研究博弈纳什均衡点的存在性,固定拓扑和时变拓扑下网络演化博弈的建模、分析及控制问题,并将所得结果应用于时尚策略决定、传染病传播和免疫控制中.主要研究内容如下：1.研究了静态博弈纯策略纳什均衡点的存在性问题.通过构建结构矩阵,将玩家的收益函数转化为代数形式.利用伪布尔函数的导数的代数形式,分别给出了完全信息和非完全信息双选择静态博弈纳什均衡点存在的充要条件.利用玩家收益函数的结构矩阵,建立了多玩家多选择静态博弈纳什均衡点的求解算法.并且针对双选择和多选择静态博弈给出了纳什均衡点存在的一个统一的充要条件.将所得的寻找纳什均衡点的结果应用到了时尚博弈纳什均衡策略选择问题中,并研究了时尚博弈中社会福利最优化问题和规范化的满意度总和最优化问题.2.研究了一类固定拓扑下网络演化博弈的代数描述和策略优化问题.为基于短视最优响应策略升级规则的网络演化博弈动态建立了构造其代数形式的算法,并分析博弈的演化结果.利用伪玩家的控制作用研究博弈的策略优化选择问题,设计控制策略使得玩家的长期平均收益最大化.3.研究了固定拓扑下网络演化博弈中玩家策略组合的稳定度问题.基于最优模仿策略升级规则,为博弈演化动态建立了代数方程.根据提出的策略组合稳定度的概念,建立了策略组合为k度稳定的充要条件,并给出了受扰策略组合还原所需要的暂态时间的计算方法.建立了一类事件触发控制的设计方法,以使得给定策略组合达到期望的稳定度.4.为时变拓扑下的网络演化博弈动态建立代数描述,并基于此分析博弈的演化规律.将所得的结果用来研究动态网络上的传染病传播问题.根据一类确定性的共演化规则,分别建立了个体状态和网络拓扑动态的矩阵表示.对于所有的初始个体状态和网络拓扑结构,分析最终可能的传播平衡点.研究了控制疾病传播的疫苗控制问题,给出了动态网络上的传染病传播能够在状态反馈疫苗控制作用下使得所有个体恢复易感状态的充分必要条件.
[Abstract]:In recent years, with the rapid development of complex network, network game has become a hot topic in game theory research. Compared with the traditional game, the network game network topology of vertices with edges indicate the presence of game player, game relationship between each adjacent game player, game player only and the neighbors for the game and get accumulated earnings this, more consistent with social reality. In biology, economics, computer science, international relations, military strategy and many other fields have been widely used, has extremely important theoretical value and significance of the research. In the limited network evolution game, each game player has a finite strategy choice, its dynamic evolution the process and logic network has a natural link. And the semi tensor product of matrices as a powerful tool to deal with a finite set of dynamic, can also be used to study the limited network evolution game The problem. Using this method can be transformed into a dynamic game algebra, a rigorous mathematical framework for it, which is conducive to the people of the game phenomenon in real life to produce more accurate and in-depth understanding, and guide people to predict and control the game. In view of the network game play diversity problem, semi tensor product the evolution of space field and the wide application of game theory in the network, it is worthy of further study. The existence of game matrix semi tensor product of Nash equilibrium, variable modeling evolutionary game under fixed topology and network topology, analysis and control problems, and the results are applied to a strategic decision, the spread of infectious diseases and immune control. The main research contents are as follows: 1. the existence problem of static game of pure strategy Nash equilibrium point. By constructing the structure matrix, the game player gain function Into algebraic form. Using algebraic form derivative pseudo Boolean functions, are given complete information and incomplete information to choose two necessary and sufficient conditions for the existence of static game Nash equilibrium point. The game player payoff function structure matrix is established, many game player selection algorithm to solve the static game Nash equilibrium. And based on the double selection and how to choose the static game to give a unified necessary and sufficient conditions for the existence of Nash equilibrium. The search for the Nash equilibrium results of the game are Nash equilibrium strategy application problems, and study the Shangbo when social welfare optimization problems and the satisfaction of the sum of the game code optimization problem is studied for a class of.2. the evolutionary game under fixed topology network algebraic description and policy optimization problem. As the game dynamic evolution myopic best response strategy to build rules based network upgrade The structure of the algebraic form of the algorithm, and analyze the evolution results of the game. The optimization selection strategy based on the pseudo control game game player, control strategy is designed to maximize the long-term average.3. game player of game playing strategy combination of stability problem of fixed topology network evolution. The optimal strategies of imitation upgrade based on the rules for the game evolution established algebraic equations. According to the proposed strategy combination stability concept, establish a strategic combination of necessary and sufficient conditions for K stability, and gives the calculation method of transient disturbance by the time required for the reduction strategy. A design method for a kind of event triggered control, so that a given strategy combination to achieve the desired stability.4. time-varying topology under dynamic network evolution game algebraic description, and based on the analysis of the evolution of the game. The results for the spread of infectious diseases research on dynamic network. According to the evolution rules of a class of deterministic, respectively established matrix individual condition and dynamic network topology. For the initial state of all individual and network topology, communication equilibrium analysis may eventually. The vaccine control is studied to control the spread of the disease and given the spread of infectious diseases on the network dynamics in state feedback control under the action of vaccine which every individual to restore the necessary and sufficient conditions for vulnerable state.

【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5;O183.2

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