以特徵向量法解條件分配相容性問題

Abstract

給定兩個隨機變數的條件機率矩陣A和B，相容性問題的主要課題包\n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\n或找出所有的聯合分配；(二)若不相容，則如何訂定測量不相容程度的方\n法並找出最近似聯合分配。目前的文獻資料有幾種解決問題的途徑，例\n如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約\n化對角塊狀矩陣法及Arnold et al. (2002)的數學規劃法等，經由這些方法\n的啟發，本文發展出創新的特徵向量法來處理前述的相容性課題。\n\n當A和B相容時，我們觀察到邊際分配分別是AB′和B′A對應特徵值1的\n特徵向量。因此，在以邊際分配檢驗相容性時，特徵向量法僅需檢驗滿足\n特徵向量條件的邊際分配，大幅度減少了檢驗的工作量。利用線性代數中\n的Perron定理和不可約化對角塊狀矩陣的概念，特徵向量法可圓滿處理相\n容性問題(一)的部份。\n\n當A和B不相容時，特徵向量法也可衍生出一個測量不相容程度的簡單\n方法。由於不同的測量方法可得到不同的最近似聯合分配，為了比較其優\n劣，本文中提出了以條件分配的偏差加上邊際分配的偏差作為評量最近似\n聯合分配的標準。特徵向量法除了可推導出最近似聯合分配的公式解外，\n經過例子的驗證，在此評量標準下特徵向量法也獲得比其他測量法更佳的\n最近似聯合分配。由是，特徵向量法也可用在處理相容性問題(二)的部份。\n\n最後，將特徵向量法實際應用在兩人零和有限賽局問題上。作業研究的\n解法是將雙方採取何種策略視為獨立，但是我們認為雙方可利用償付值表\n所提供的資訊作為決策的依據，並將雙方的策略寫成兩個條件機率矩陣，\n則賽局問題被轉換為相容性問題。我們可用廣義相容的概念對賽局的解進\n行分析，並在各種測度下討論賽局的解及雙方的最佳策略。

Given two conditional probability matrices A and B of two random\nvariables, the issues of the compatibility include: (a) how to determine\nwhether they are compatible? If compatible, how to check the uniqueness\nof the joint distribution or find all possible joint distributions; (b)\nif incompatible, how to measure how far they are from compatibility\nand find the most nearly compatible joint distribution. There are\nseveral approaches to solve these problems, such as the ratio matrix\nmethod(Arnold and Press, 1989), the IBD matrix method(Song et\nal., 2010) and the mathematical programming method(Arnold et al.,\n2002). Inspired by these methods, the thesis develops the eigenvector\napproach to deal with the compatibility issues.\n\nWhen A and B are compatible, it is observed that the marginal distributions\nare eigenvectors of AB′ and B′A corresponding to 1, respectively.\nWhile checking compatibility by the marginal distributions, the\neigenvector approach only checks the marginal distributions which are\neigenvectors of AB′ and B′A. It significantly reduces the workload.\nBy using Perron theorem and the concept of the IBD matrix, the part\n(a) of compatibility issues can be dealt with the eigenvector approach.\n\nWhen A and B are incompatible, a simple way to measure the degree\nof incompatibility can be derived from the eigenvector approach.\nIn order to compare the most nearly compatible joint distributions\ngiven by different measures, the thesis proposes the deviation of the\nconditional distributions plus the deviation of the marginal distributions\nas the most nearly compatible joint distribution assessment standard.\nThe eigenvector approach not only derives formula for the most\nnearly compatible distribution, but also provides better joint distribution \nthan those given by the other measures through the validations\nunder this standard. The part (b) of compatibility issues can also be\ndealt with the eigenvector approach.\n\nFinally, the eigenvector approach is used in solving game problems.\nIn operations research, strategies adopted by both players are assumed\nto be independent. However, this independent assumption may not\nbe appropriate, since both players can make decisions through the\ninformation provided by the payoffs for the game. Let strategies of\nboth players form two conditional probability matrices, then the game\nproblems can be converted into compatibility issues. We can use the\nconcept of generalized compatibility to analyze game solutions and\ndiscuss the best strategies for both players in a variety of measurements.