基于半张量积方法的有限博弈的结构、演化稳定性与控制

发布时间：2017-12-26 20:39

本文关键词：基于半张量积方法的有限博弈的结构、演化稳定性与控制　出处：《山东大学》2017年博士论文　论文类型：学位论文

【摘要】：演化博弈论是博弈论研究中的一个新领域,是在博弈论的基础上发展起来的一种理论.演化博弈论研究有限理性的个体如何随着时间的推移在不断重复的博弈过程中,通过一定的学习规则升级自身的策略,从而使得收益最大化.目前,演化博弈论在生物学、经济学、社会学、工程科学和其他诸多领域都得到了广泛的应用,因此有着极其重要的理论价值和研究意义.在演化博弈论中,当参与人的个数和策略的个数都有限时,其演化过程就是一个有限值的逻辑动态过程,有限演化博弈可以等价为一个有限值逻辑动态系统.矩阵半张量积方法已被成功地应用于逻辑动态系统的分析和控制中,因此该方法同样可以用来研究有限演化博弈问题.本论文利用矩阵半张量积方法,首先研究了有限非合作博弈的加权子空间分解,然后考虑了马尔科夫型演化博弈的稳定性与镇定问题,最后讨论了时滞演化博弈的建模及稳定性.主要研究内容如下:1.研究了有限非合作博弈的加权子空间分解.首先,通过分析有限非合作博弈的性质,确定了它的向量空间结构,并将其正交分解为三个加权子空间的和,这三个加权子空间分别是加权纯势博弈子空间、非策略博弈子空间和加权纯调和博弈子空间.其次,给出了各个加权子空间的基底,从而为我们研究它们的性质提供了便利工具.最后,利用这些基底,构造了加权基矩阵,得到了加权子空间的计算公式和一些性质,同时考虑了权值优化问题.2.基于李雅普诺夫函数的方法,研究了马尔科夫型演化博弈的稳定性和镇定问题.首先,通过定义k-值逻辑动态系统的李雅普诺夫函数,得到了马尔科夫型纯策略演化博弈全局稳定的充分必要条件,并通过解一个不等式组,给出了李雅普诺夫函数的构造算法.其次,对于时变收益函数的演化博弈,构造了共同李雅普诺夫函数以保证全局稳定性.进一步利用势方程,建立了一个基于势函数的共同李雅普诺夫函数,得到了近似势博弈演化稳定的条件.最后,基于稳定性分析的结果,讨论了马尔科夫型控制演化博弈的镇定问题,给出了可全局镇定的一些条件.3.研究了时滞作用对演化稳定性的影响.首先,利用矩阵半张量积,分别给出了时不变时滞和时变时滞演化博弈的代数描述,并通过一定的转换,将时滞演化博弈转化为不带时滞的标准离散动态系统.其次,考虑了基于势博弈的时滞演化动力学,通过设计具体的策略更新规则,保证时滞演化博弈收敛到一个纯纳什均衡.最后,研究了基于势博弈的时滞网络演化博弈的稳定性。
[Abstract]:Evolutionary game theory game theory is a new field of research, is a theory developed based on game theory. Evolutionary game theory study the limited rationality of how individuals with time in the process of the game is repeated in the upgrading of self strategy by learning the rules of making maximum profit at present, the evolutionary game theory in Biology, economics, sociology, engineering science and many other fields have been widely used, so it has extremely important theoretical value and significance. In the evolutionary game theory, and when the number and the number of strategies is limited, its evolution is a finite value logic dynamic process, can be co evolutionary game equivalent to a finite value logic dynamic system. Analysis and control matrix semi tensor product method has been successfully applied in dynamic logic system This system, so this method can also be used to study the co evolutionary game problem. This paper uses semi tensor product matrix, firstly, non cooperative game weighted subspace decomposition, and then consider the Markov type evolutionary game stability and stabilization problems are discussed, the modeling and stability of the game evolution lag. The main research contents are as follows: 1. research on non cooperative game weighted subspace decomposition. Firstly, through the analysis of the nature of non cooperative game, the vector space of its structure, and the orthogonal decomposition into three weighted subspace and the three weighted subspace are game subspace and weighted non weighted pure potential game space, pure harmonic game space. Secondly, the base has given each weighted subspace, which provides a convenient tool for us to study their properties. Finally, The use of these substrates, construct the weighted matrix, the calculation formula of weighted subspace and some properties, and consider the method of Lyapunov function based on the optimal weights of.2., and stability of Markovian evolutionary game. First of all, the definition of k- value system Lyapunov function get the necessary and sufficient conditions for a Markov type pure strategy evolutionary game of global stability, and by solving a group of inequalities, gives the algorithm to construct the Lyapunov function. Secondly, the evolutionary game for the time-varying gain function, construct the common Lyapunov function to guarantee global stability. Further use of potential equations, established a a common Lyapunov function and potential function based on the approximate potential game evolutionary stable conditions. Finally, the results of stability analysis based on the discussion of the horse The stabilization problem of evolutionary game Erkefu type control, given some conditions of global.3. stabilization effect of the time-delay effect on evolutionary stability. First, by using the semi tensor product of matrices, are given algebraic description for time invariant delays and time-varying delays of evolutionary game theory, and through the conversion, the delay is transformed into evolutionary game with standard discrete dynamic system. Secondly, considering the dynamic evolution of delay potential game based on the update rules through specific design strategies, which delay evolutionary game converges to a pure Nash equilibrium. Finally, stability of evolutionary game based on the network delay potential game.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O225

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