【作者】 方志耕；

【导师】 刘思峰；

【作者基本信息】 南京航空航天大学 ， 管理科学与工程， 2007， 博士

【摘要】 在博弈论中,除了不完全信息和有限理性等之外,还有未来的不确定性、有限知识(或称有限信息、贫信息[5])等等许多问题。然而按照目前学术界所惯用的对博弈问题从信息角度的划分来看,其所谓信息的完全与不完全主要是指博弈参与人的信息对称与不对称,在一定程度上可以说,它存在着忽略了信息“缺失”等不确定性问题研究的较严重缺陷。事实上,由于各种随机因素和非随机因素的影响,既使在较严格的限制条件下,使得现实中的这种任意两次博弈的损益值也不可能完全一致。也就是说,在现实中,这种博弈的损益值不可能是完全清楚和准确的,经典博弈理论所要求的分析条件难以得到满足,存在着信息“缺失”(或称有限知识)问题,这种信息“缺失”问题普遍存在[1,2,3]。人们对系统的认识不可能都是十分完全的,展现在人们面前的系统往往不是“白”的,而是“灰”的,博弈理论中所涉及到的许多问题几乎都是灰的。然而,目前的经典博弈理论有关信息“缺失”问题的研究极少,对现实中的灰系统几乎都采用了过份简化的方法(将这些“灰系统”简单地看作“白系统”)进行处理,其结果导致了博弈论的预测对现实的指导作用大打折扣。我的博士论文选题《灰色博弈理论及其经济应用研究》,就是想借用灰系统理论的丰富理论与相关的方法手段来研究和解决博弈论中的有限理性和有限知识等问题。创建与现实经济问题结合更加紧密的灰矩阵、灰双矩阵和灰进化博弈模型,并设计出简洁、高效的解的概念及其结构体系,完成其一些基础性的研究工作,创建与设计灰矩阵、灰双矩阵和灰进化博弈理论的框架(体系),从而为解决现实经济问题提供一种有力的工具――灰矩阵、灰双矩阵和灰进化博弈理论。论文首先提出了这种信息缺失的灰博弈问题,并运用灰色系统理论对这种博弈问题进行了定义(称其为灰博弈问题),构建了基于纯策略和混合策略的灰矩阵博弈模型,提出并建立灰鞍点和灰混合策略解的概念及其结构体系;主要解决了由于原灰数运算体系的缺陷所造成的灰矩阵博弈混合策略解求解的困难(初步研究显示,原灰数运算方法很容易将灰混合策略解不正常的放大为毫无意义的黑数)问题;给出了该博弈问题纯策略意义下的“灰鞍点”解的表征形式与定义;并研究了其相关求解算法;证明对于任何灰矩阵博弈问题,一定存在灰数意义下的解(即,最大最小灰值定理)。论文定义了最乐观和最悲观值矩阵、灰优和灰劣策略、虚增策略和虚增收益值向量、长行维和长列维灰矩阵、灰满秩扩充方阵及其灰逆阵等一系列重要概念;证明了局中人的虚增策略是原博弈问题的最劣等策略、虚拟灰损益值方阵的构造、局中人的最优灰博弈策略和灰博弈值、灰满秩扩充方阵存在的充要条件、灰降秩扩充方阵的满秩化处理、满秩化处理不改变原G ( ?)的最优解等一系列的定理。最终,解决了基于非满秩灰损益值矩阵的矩阵博弈的矩阵法求解问题,建立了求解灰矩阵博弈混合策略解的简便、实用和高效的计算方法。论文首先引入并解决了由于灰矩阵博弈问题的信息缺失所造成的最优灰矩阵博弈混合解的风险表征、测度及其控制问题;建立和完善了灰矩阵博弈的风险测度与控制的理论体系。本论文基本构架和完成了灰色矩阵博弈理论。论文首次提出了一类博弈信息缺损条件下的博弈均衡分析及其均衡点的存在性问题;建立了损益值信息对称缺损的静态博弈的框架,收益值区间灰数的均势度、优势度、劣势度,灰势—纯策略纳什均衡,灰势—纯策略上策均衡等概念和区间灰数势关系的势判定规则;证明了:如果某损益值信息对称缺损的n人静态博弈问题存在灰势—纯策略纳什均衡的话,那么视不同情况,运用灰势—纯策略上、下策分析法,灰势—划线、箭头分析法能够方便地寻找到该均衡,从而该类博弈问题可以做出较为可信的预测。论文以某一现实的损益值信息对称缺损的彩电价格竞争的静态博弈问题为背景,研究了其灰势纳什均衡点及其灰势优胜策略,对现实具有较好的解释力。动态博弈分析的中心内容是子博弈完美纳什均衡分析,子博弈完美纳什均衡分析的核心方法是逆推归纳法。长期以来,逆推归纳法悖论与现实严重不符的现象困扰着学术界。论文揭示了逆推归纳悖论产生的根源:首先是其所犯的微观逻辑推理对整体宏观逻辑观忽略的错误;或者说只重视眼前(近期)利益,而忽略长远利益;其次是经典的多阶段动态博弈模型的结构形式无法满足人们对整体的和长远的利益考虑与均衡分析。本文构建了一种新型的基于未来博弈引导值的动态博弈模型的结构形式;设计了多阶段动态博弈的逆推“灰数规整”算法;构建多阶段动态博弈的“终止”和“引导”纳什均衡解的概念体系,并提供了方便有效的均衡分析方法;从而较好地破解了“蜈蚣博弈”的悖论。论文针对目前的进化博弈模型不能对一次性博弈结果或短期经济均衡等进行预测的缺陷,构建了基于对称和非对称情形的进化博弈的博弈链模型。深刻地揭示了,博弈各方在进化博弈过程中的相互依存与相互转换的关系,建立了博弈各方在每一步博弈过程中的个体数量及其期望平均收益的递推关系。在此基础之上,我们以某鹰-鸽博弈为例,对其复制动态与进化稳定策略进行了仿真分析。仿真实验表明:在对称情形的鹰-鸽博弈过程中,存在唯一的进化稳定策略的均衡点x,在该点的左边区域是鹰的复制进化区域(鹰的数量增长,鸽的数量减少),而在该点的右边区域是鸽的复制进化区域(鸽的数量增长,鹰的数量减少);根据非对称鹰-鸽博弈仿真实验,我们能够判定:博弈方1和2的鹰的复制进化区域(鹰的数量增长,鸽的数量减少)、临界初始值曲线和进化稳定策略所决定的均衡点。通过仿真实验,论文首次揭示了生物的试错进化博弈现象。论文运用灰系统理论的思想,对目前的一级密封价格拍卖博弈模型进行检验和验证,并对其存在的一些缺陷进行了剖析,认为这些经典模型对条件的限制过于严格,与现实的吻合性较差.基于有限理性假设,设计了经验理想报价灰修正系数,建立了基于准确的价值和经验理想报价估价的有限理性最优灰报价模型。对该模型灰系数进行第一标准灰数变换,找到了投标人的威胁反映灰系数;发现了投标人的最优灰报价不仅取决于其自身的价值,而且还取决于他人的价值及其威胁反映灰系数;投标人的最优灰报价不仅仅刚好为其对被拍物品所认可价值的一半,而要视情而定,一般情况下均高于其所认可价值的一半。对该模型进行了数据仿真,得到一些与经典模型有较大差异的有价值的结论,并建议了投标人的最佳投标模式。更多还原

【Abstract】 In Game theory, there are many problems, such as incomplete information, bounded rationality and uncertainty of future (or poor information, Grey System and its Application, Sifeng Liu, 1999). However nowadays in the view of information in game theory, the complete and incomplete information means respectively symmetry and asymmetry information of players, and to a certain extent, there exits a serious defect of omitting the uncertainty of information such as lack of information. In fact, because of all sorts of stochastic and non-stochastic factors,the pay-offs of random twice games could not be consistent though it is on strict restricted condition. In other words, the pay-off matrix of game can not be clear and exact, so that the analysis condition of classical game theory is hard to be satisfied and lack of information and finite knowledge is ubiquitous [1-3]The understanding of people to system is impossible to be absolutely complete for that the system people face usually is not“white”but“grey”. Here use“white”to represent completely known information, and“grey”for those information which are is partially known and partially unknown. Many problems discussed in Game Theory almost are grey. But in classical game theory, the grey systems of real life are always dealt with by white system by devilishly predigestion, with the resultant that the guidance of forecast by game theory largely decreases.The project of my dissertation, Research on the Grey Game Theory and Its Application in Economy, aims to study the problems of bounded rationality and incomplete information of Game Theory by the abundant theory and correlative methods of Grey System. The game models of grey matrix, grey bi-matrix and grey evolutionary game, which could describe economic problems better and more truly, are put forward, and concise and effective conceptions and structure of solution are worked out. By some foundation works of system of grey matrix, grey bi-matrix and grey evolutionary game, this dissertation presents a powerful tool to know and settle real economic problem.Above all, the dissertation discovers the game problem with incomplete information, which is defined as grey game problem by grey system theory, and advances the Grey Matrix Game Model based on pure strategy and mixed strategy. The main works are setting up the conceptions and structure of grey saddle and grey mixed strategy, and settling the difficulty of finding grey mixed strategy of grey matrix game due to the original systematic defect of grey number computation. It has been found that, mixed strategy solution is easy to be magnified to useless black number by present method. The definition and form of grey saddle of pure strategy of game problem are provided in the dissertation, and also arithmetic of solution is researched. And it proves that random grey matrix game must have a solution of grey form, which is given in maximum minimize theorem.In this dissertation, a series of important conceptions are advanced, such as conceptions of the most optimistic and pessimistic value matrix, superiority and inferior strategy, long row grey matrix, long column grey matrix, grey full rank extended square matrix and its grey inverse matrix. Moreover a series of theorems are proved, including theorem of virtual added strategy of player being the most inferior strategy of original game, theorem of constitution of virtual grey pay-off square matrix, and also including theorem of the optimal strategy and grey value of player, the sufficient and necessary condition of grey full rank extended square matrix, theorem of full rank process of grey singular extended square matrix and theorem that full rank process does not change the optimal solution of original game. On the basis of these works, author found the solution to non-full rank grey pay-off matrix game, and gave the simple, practical and effective solution to mixed strategy of grey matrix game. For the first time, the dissertation puts forward and solutes the denotation, measure and control problems of overrated and underrated risks of optimal mixed strategy, which are due to the incomplete information of grey matrix game. Furthermore the theoretic system of measure and control of risk of grey matrix game is set up and developed. And here the main structure and theory of grey matrix game is mainly completed on the whole.For the first time, the dissertation advanced the equilibrium analysis problem and existence problem of equilibrium point in the condition of incomplete information. And the static game structure with revenue of symmetric information loss is built up, and a series of conceptions are developed, including superiority, inferior and equipollence position degrees of income in interval grey number, grey position-pure strategy Nash equilibrium, and grey position-pure dominant strategy equilibrium, also decision rules of grey positions of grey interval numbers are presented. It is proved that, if the grey position-pure strategy Nash equilibrium exists in n -players static game with revenue of symmetric information loss, it could be easily found according to different conditions by the analysis methods of grey position-pure dominant strategy, grey position-pure dominated strategy and grey position-underline-arrow, therefore the result of this category game could be forecasted quite reliably. Taking the example of price competition of color TV set, whose revenue is in symmetric information loss in real life, thesis discusses its grey position-pure strategy Nash equilibrium point and its grey position- dominant strategy, which explain real life effectively.The core study of dynamic game is perfect Nash equilibrium analysis of subgame, the key method of which is converse inductive method[8,9]. For a long time, the situation of paradox of converse inductive method always puzzled academia. This thesis discovers the root of paradox of converse inductive method, and it has two aspects. On one hand, during using this method, the microscopically logical illation makes the macroscopically logical viewpoint omitted, and in other words, emphasizing short-term interest but neglecting long-term interest. On the other hand, the classical model structure of multi-phase dynamic game could not be used to analyze the whole and long-term interest. This thesis builds up a new model structure of dynamic game based on pilot value of future income, designs the arithmetic of regularization of grey number and the conception system of the end and the pilot of Nash equilibrium solution, and also provides the effective equilibrium analysis method. All these works help solute the Paradox of Centipede Game well.The dissertation overcomes lacuna of evolutionary game model that the model can not forecast results of one-off game and short-term economic equilibrium, and designs evolutionary game chain model based on symmetry and asymmetry cases. Furthermore, it discovers profoundly relationships that players in the game are in dependence and conversion each another and establishes transfer formulas of player quantities and expectation average payments in per step of the game. On the basis of that, taking the eagle-pigeon game as an example, author imitates and analyzes copy dynamics and evolutionarily stable strategies (ESS). Imitated experiment shows the conclusion that there is exclusive equilibrium point of ESS x on symmetry case of the game, the left area of which is copy evolutionary area of the eagle that quantity of the eagle increase and quantity of the pigeon decrease in process of the game, and the right area of which is copy evolutionary area of the pigeon that quantity of the pigeon increase and quantity of the eagle decrease in process of the game. On asymmetry case, the equilibrium point, which is decided by copy evolutionary areas of the eagle and pigeon, curves of initial critical value and evolutionarily stable strategies, could be gotten by imitation. This thesis firstly discovers the biology evolutionary phenomenon of trial and error.It is found that there are some defects in the classical first-price sealed auction model, whose conditions are restricted too much to fit the real situation, by thoughts of grey system. The dissertation designs grey correction factor of experiential ideal quotation, and builds optimal grey quotation model based on accurate evaluation of value and experiential ideal quotation based on finite reasonability. However, bidder’s menace reflection coefficient is gotten by the way of first standard grey transform. It is discovered that bidder’s optimal grey quotation depends not only on values the bidder itself estimates but on values the rival estimates, and also on menace reflection grey coefficient. The optimal grey quotation of bidder is not the half of values of goods at auction, but higher than the half of the values generally. Furthermore, with simulation of the model, the dissertation proposes significantly some conclusions which are different from the classical model, and the optimal patterns of bidding.更多还原