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    题名: 特徵向量法在三維條件分配相容性問題上之研究
    On the compatibility issues of three-dimensional conditional distributions by eigenvector approach
    作者: 高裕哲
    Kao, Yu Che
    贡献者: 宋傳欽
    Sung, Chuan Chin
    高裕哲
    Kao, Yu Che
    关键词: 條件分配
    相容性
    特徵向量法
    三人賽局
    單純策略
    混合策略
    納許均衡策略
    完全均衡策略
    合適均衡策略
    conditional distributions
    compatibility
    eigenvector approach
    three-person game
    pure strategy
    mixed strategy
    Nash equilibrium
    perfect equilibrium
    proper equilibrium
    日期: 2012
    上传时间: 2013-09-02 16:46:08 (UTC+8)
    摘要: 給定一些隨機變數的條件分配，一般相容性問題的研究包含：(一)如何判斷他們是否相容？若相容，則如何檢驗聯合分配的唯一性或找出所有的聯合分配；(二)若不相容，則如何訂定評量近似聯合分配的標準並盡可能找出好的近似聯合分配。
    顧仲航(2011)提出了二維特徵向量法解條件分配相容性問題並實際應用在兩人零和賽局上。本文中，我們嘗試將二維特徵向量法擴展至三維上並應用到解三人賽局的問題。
    特徵向量法在三維中較複雜，因此本文將給定的條件分配簡單分成對稱型及不對稱型兩類。若條件分配為對稱型，則可以二維特徵向量法的技巧來處理相容性問題，對於處理的過程我們提供了詳細的步驟。若條件分配為不對稱型，則通常無法獲得一般的處理流程，必須針對不同的狀況採取不同的方式來應對。在某些條件分配的組合下，我們用實例說明特徵向量法仍可用來處理相容性問題。當給定的條件分配不相容時，我們也提出了三維中近似聯合分配的求法。
    最後，將特徵向量法應用在三人賽局問題上。作業研究中的解法是假設三位參賽者的策略選擇為獨立，但我們認為三位參賽者可由償付值表所提供的資訊作為策略選擇的依據，在決策上彼此是不獨立的。從償付值表經常可獲得和三位參賽者決策有關的對稱型條件分配，賽局問題被轉換為相容性問題，進而可依處理相容性問題的過程求賽局的解。我們也以實例說明，當賽局有多重均衡解時，三維特徵向量法可從償付值表所提供的部分訊息進一步求得最適合的均衡策略。
    Given a set of conditional distributions of random variables, the compatibility issues include: (1) how to determine whether they are compatible? If compatible, how to check the uniqueness of the joint distribution or to find all possible joint distributions; (2) if incompatible, how to set standards for evaluating near joint distributions and to find a good one.
    Ku(2011) proposes a two-dimensional eigenvector approach to solve compatibility issues and applies it to two-person zero-sum game. In this paper, we try to extend the eigenvector approach to the three-dimensional case and apply it to solve three-person game problems.
    Eigenvector approach is more complex in three-dimension than in two-dimension, so we simply classify the given conditional distributions into two types, symmetric and asymmetric. When the conditional distributions are symmetric, we may solve them by using the same skills in two-dimensional eigenvector approach. Detailed steps for the process are provided. When the conditional distributions are asymmetric, we usually deal with them case by case. For some special asymmetric conditional distributions, several examples are given to demonstrate that eigenvector approach still works. When the given conditional distributions are incompatible, a method for finding near joint distributions is also given.
    Finally, the eigenvector approach is used in solving three-person game problems. In operations research, players are assumed to adopt strategies independently. However, this assumption is inappropriate, since players can make their decisions through the information provided by the payoffs for the game. Frequently, a set of symmetric conditional distributions can be derived from the given payoffs. The game problems are then converted into compatibility issues and can be addressed by the results of compatibility theory. We also use an example to show that our three-dimensional eigenvector approach can distinguish the most appropriate equilibrium strategy from the others through additional information given by the payoffs when the game has multiple equilibriums.
    參考文獻: [1] Arnold, B. C., Castillo, E., amd Sarabia, J. M. (2002), Exact and near compatibility of descrete conditional distributions. Computational Statistics & Data Analysis, 40, 231-252.
    [2] Kuo, K. L. (2008), New tools for studying the Fergyson-Dirichlet process and compability of a family of conditionals.，政治大學應用數學系博士論文。
    [3] Myerson, R. B. (1977), Refinements of the Nash Equilibrium Concept. Int. Journal of Game Theory, Vol. 7, Issue 2, 73-80.
    [4] Song, C. C., Li, L. A., Chen, C. H., Jiang, T. J., and Kuo, K. L. (2010), Compatibility of finite descrete conditional distributions. Statistical Sinica, 20, 423-440.
    [5] 謝淑怡 (1995)，賽局理論，三民書局，台北市。
    [6] 藍兆杰、徐偉傑、陳怡君 (譯) (2002)，策略的賽局 (原作者：Avinash Dixit & Susan Skeath)，弘智文化，台北市。
    [7] 顧仲航 (2011)，以特徵向量法解條件分配相容性問題，政治大學應用數學系碩士論文。
    描述: 碩士
    國立政治大學
    應用數學研究所
    99751009
    101
    資料來源: http://thesis.lib.nccu.edu.tw/record/#G0099751009
    数据类型: thesis
    显示于类别:[應用數學系] 學位論文

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