給定聯合分配，可以容易地導出對應的條件分配。反之，給定條件分配的資訊，是否能導出對應的聯合分配呢？例如根據O. Paul et al.（1963，1968）對造成心血管疾病因素之追蹤研究，可得出咖啡量、吸菸量及是否有心血管疾病三者間的條件機率模型資料，是否能找到對應的聯合機率模型，以便可以更深入地研究三者之關係，是一個重要的議題。在選定參考點下，Chen（2010）提出以勝算比法找條件密度函數相容的充要條件，以及在相容性成立時，如何求得聯合分配。在二維中，當兩正值條件機率矩陣不相容時，郭俊佑（2013）以幾何平均法修正勝算比矩陣，並導出近似聯合分配，同時利用幾何平均法之特性，提出最佳參考點之選擇法則。本研究以二維的勝算比法為基礎，探討三維離散的相容性問題，獲得下列幾項結果：一、證明了三個三維條件機率矩陣相容的充要條件就是兩兩相容。二、當三維條件機率矩陣不相容時，利用幾何平均法導出近似聯合分配。三、利用兩兩相容的充要條件，導出三維條件機率矩陣相容的充要條件，並證明該充要條件與Chen的結果一致。四、在幾何平均法下，提出最少點法，有效率地找出最佳參考點，以產生總誤差最小的近似聯合分配。五、設計出程式檢驗三維條件機率矩陣是否相容，並找出最佳參考點，同時比較最少點法與窮舉法之間效率的差異。 / Given a joint distribution, we can easily derive the corresponding fully conditional distributions. Conversely, given fully conditional distributions, can we find out the corresponding joint distribution? For example, according to a longitudinal study of coronary heart disease risk factors by O. Paul et al. (1963, 1968), we obtain conditional probability model data among coffee intake, the number of cigarettes smoked and whether he/she has coronary heart disease or not. Whether we can find out the corresponding joint distribution is an important issue as the joint distribution may be used to do further analyses. Chen (2010) used odds ratio method to find a necessary and sufficient condition for their compatibility and also gave the corresponding joint distribution for compatible situations. When two positive discrete conditional distributions in two dimensions are incompatible, Kuo (2013) used a geometric mean method to modify odds ratio matrices and derived an approximate joint distribution. Kuo also provided a rule to find the best reference point when the geometric mean method is used. In this research, based on odds ratio method in two dimensions, we discuss their compatibility problems and obtain the following results on three-dimensional discrete cases. Firstly, we prove that a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions is pairwise compatible. Secondly, we extend Kuo’s method on two-dimensional cases to derive three-dimensional approximate joint distributions for incompatible situations. Thirdly, we derive a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions in terms of pairwise compatibility and also prove that this condition is consistent with Chen’s results. Fourthly, we provide a minimum-points method to efficiently find the best reference point and yield an approximate joint distribution such that total error is the smallest. Fifthly, we design a computer program to run three-dimensional discrete conditional probability matrices problems for compatibility and also compare the efficiency between minimum-points method and exhausting method.

台北市自民國72年起，公佈實施土地使用分區管制規則以來，至今已十幾年，但對於住宅區之環境品質並無明顯改善，反而突顯實施土地使用分區之分組及允許使用組別的問題，究其原因，是法規制定太偏於混合使用及允許使用組別太多。尤其附條件允許組別更為嚴重，幾乎與商業區差不多，而其中產生一些行業別與住宅不相容的情形。 有鑑於此，本研究乃期望透過混合使用相關研究回顧遴選出適當之評估指標項目，再收集專家學者對評估指標之意見，進行修正所遴選之評估指標，作為混合使用環境品質之評估基礎。在其中發現混合使用評估指標相對權重以「守望相助與警衛」權重最大、其次是「購物方便性」、第三為「餐飲方便性」。而「垃圾收集」與「污水排放問題」之小指標相對權重較小。由此不難理解，在專家的觀念裡混合使用之居住環境需以居住安全性之「守望相助與警衛」為第一考量，然後才能考慮混合使用生活之便利性。 由以上所建構之評估指標，經收集居民的意見與專家之賦予權重之後，經整理計算後，所得到評估指數正負得知，混合使用環境品質評估指標中以「寧便居住環境」、「公共安全」之指數呈現正值，而「環境污染公害」為負值，將其大指標之指數加總，發現總值為正數，得知混合使用之居住環境可能是好處多於壞處。 得知評估指標之指數大小之後，筆者再建立相容性行業別操作方式，進行計算行業別之評估指數大小，並建立數學模式，發現行業別之指數總和越大，代表居民與專家愈能接受該行業。負面值越大表示排斥性越大。同時可理解相容性越高之行業對環境影響越少，且對生活機能幫助越大。進而可透過行業別之排序，依混合使用環境品質需求，再訂定不同混合標準，以塑造不同等級之居住環境。 另在新、舊地區之實證過程中得知；混合使用的行業會因道路寬度、區位而有不同混合比例，其中立體混合也與平面混合成正相關。而從實際新、舊地區之業種調查發現，在大安區之業種是屬於都市型，而在萬華區之業種是屬於鄰里型。 綜合言之；本研究透過相關理論回顧與分析，所建立住宅區土地混合使用之評估指標，作為相容性之行業別判斷依據。在其中發現行業別之相容性，可依本研究所擬之評估指標判斷業別對居住環境的影響情形，同時可透指數分析再訂定業種相容性標準，而進行衡量評估指標與行業別之相關性計算。尤其在行業別設立時，將可能對住宅使用產生正面或負面影響情形，以指標的方式表達；如行業別是否產生噪音振動、垃圾處理方式、停車問題、空氣污染..等等問題，儘可能再設有評估標準。待相容性行業別執行一段時間後，依執行情形來修正評估標準，以利後續土地混合使用業種相容性之參考。 / Since the announcement and execution of the Zoning Act of Taipei Metropolitan in 1983, there is no obvious improvement on the living quality in the residential areas. Conversely, there are some issues on the usage classification and permitted usage categories. After further research, it’s found that the Zoning Act biased on the mixed-use, which permitted too much usage in one category. The most serious one was the permitted usage in its appendix, which made the usage of residential area almost the same as the commercial area. After researching on the related documents on mixed-use, the study emerged the appropriate evaluation indicators. Then collecting the experts’ comments, the study modified evaluation indicators accordingly, which formed the bases of environmental evaluation of mixed-use. In between, the relative importance of top three indicators was ranked as “residents’ mutual help and guard”, “shopping convenience” and “dinning convenience”. Both “trash collection” and “waste water manipulation” were less concerned. It understandable that in experts’ concepts, the residential safety was the first consideration then the living convenience. Based on the above evaluation indicators and collected comments from the residents and experts, the study came out the positive and negative values of evaluation indicators. The “quiescence and convenience of living environment” and “public safety” were positive value but “environmental pollution” was negative value. Summing up the big indicators, the value was positive which meant people possibly evaluated highly on the mixed-use in the residential environment. Understanding the importance of indicators, the study rebuilt the operation model for compatible businesses, which calculated the values of business evaluation indicators and built mathematical model. Then the study found out that the greater positive values of summed up indicators, the more representative residents and experts accepted the business. The greater the negative value, the more exclusivity represented. More compatible businesses would less impact the environment but enhance the living functionality. Ranking the business sequences according to the needs of environmental quality in mixed-use, we could build up different grades of living environments by setting up different mixed-use standards. It was proven from the study on the new and old residential areas that the mixed-use ratio of businesses would be differed by the width of road and location in which vertical and horizontal mixed-use are positively related. From the field study, the businesses in Ta-An district belonged to the metropolitan type but those in Wan-Hua district belonged to the tithing type. In summary, the study went through the review and analysis of related thesis then built the evaluation indicators for mixed-use in residential area, which in turn were the bases to identify the businesses compatibility. Utilizing the evaluation indicators, we could identify the business compatibility and its impact on the living environment. Simultaneously, through indicator analysis, new business compatibility could be redefined and evaluate the relationship between evaluation indicators and business relativity. Especially for the business category, which would heavily impact the environment, the indicators could be identified as, e.g., voice and vibration pollution, trash manipulation, parking issue, air pollution...which needed more evaluation standards. After some time of execution, the results could be used to modify the evaluation standard, which would be the reference for the succeeding study of land mixed-use.

中文摘要 有限離散條件分配相容性問題可依相容性檢驗、唯一性檢驗以及找出所有的聯合機率分配三層次來討論。目前的文獻資料有幾種研究方法，本文僅分析、比較其中的比值矩陣法和圖形法。 二維中，我們發現簡化二分圖的分支與IBD矩陣中的對角塊狀矩陣有密切的對應關係。在檢驗相容性時，圖形法只需檢驗簡化二分圖中的每個分支，正如同比值矩陣法只需檢驗IBD矩陣中的每一個對角塊狀矩陣即可。在檢驗唯一性時，圖形法只需檢驗簡化二分圖中的分支數是否唯一，正如同比值矩陣法只需檢驗IBD矩陣中的對角塊狀數是否唯一即可。在求所有的聯合機率分配時，運用比值矩陣法可推算出所有的聯合機率分配，但是圖形法則無法求出。 三維中，本文提出了修正比值矩陣法，將比值數組按照某種索引方式在平面上有規則地呈現，可降低所需處理矩陣的大小。此外，我們也發現修正比值矩陣中的橫直縱迴路和簡化二分圖中的迴路有對應的關係，因此可觀察出兩種方法所獲致某些結論的關聯性。在檢驗唯一性時，圖形法是檢驗簡化二分圖中的分支數是否唯一，而修正比值矩陣法是檢驗兩個修正比值矩陣是否分別有唯一的GROPE矩陣。修正比值矩陣法可推算出所有的聯合機率分配。 圖形法可用於任何維度中，修正比值矩陣法也可推廣到任何維度中，但在應用上，修正比值矩陣法比圖形法較為可行。 / The issue of the compatibility of finite discrete conditional distributions could be discussed hierarchically according to the compatibility, the uniqueness, and finding all possible joint probability distributions. There are several published methods, but only the Ratio Matrix Method and the Graphical Method are analyzed and compared in this thesis. In bivariate case, a close correspondence between the components of the reduced bipartite graph and the diagonal block matrices of the IBD matrix can be found. When we examine the compatibility, just as simply each diagonal block matrix of the IBD matrix needs to be examined using the Ratio Matrix Method, so does each component of the reduced bipartite graph using the Graphical Method. When we examine the uniqueness, just as whether the number of the diagonal blocks of the IBD matrix is unique needs to be examined, so does the number of the components of the reduced bipartite graph. The Ratio Matrix Method can provide all possible joint probability distributions, but the Graphical Method cannot. In trivariate case, this thesis proposes a Revised Ratio Matrix Method, in which we can present the ratio array regularly in the plane according to the index and reduce the corresponding matrix size. It is also found that each circuit in the revised ratio matrix corresponds to a circuit in the reduced bipartite graph. Therefore, the relation between the results of the two methods can be observed. When we examine the uniqueness with the Graphical Method, we examine whether the number of the components in the reduced bipartite graph is unique. But with the Revised Ratio Matrix Method, we examine whether each revised ratio matrix has a unique GROPE matrix. All possible joint probability distributions can be derived through the Revised Ratio Matrix Method. The Graphical Method can be applied to the higher dimensional cases, so can the Revised Ratio Matrix Method. But the Revised Ratio Matrix Method is more feasible than the Graphical Method in application.

本文考慮當資訊不對稱下的逆向選擇問題存在時,如何遵循Myerson(1979)提出的揭露原則 (the revelation principle)來設計一套具備誘因相容性 (incentive compatibility)的存款保險契約,契約中的自有資本比率為要保機構的自我選擇變數 (self-selection variable),而保險費則為存保公司用來控制要保機構決策行為的控制變數(control var iable),依此可以建立一套自我選擇機能(self-selection mechanism),來促使要保機構誠實揭其風險類型的私有訊息(private information),進而將要保機構依風險高低正確分類,徹底解決訊息不對稱下的逆向選擇問題。

設二個隨機變數X1 和X2，所可能的發生值分別為{1,…,I}和{1,…,J}。條件機率分配模型為二個I × J 的矩陣A 和B，分別代表在X2 給定的條件下X1的條件機率分配和在X1 給定的條件下X2的條件機率分配。若存在一個聯合機率分配，而且它的二個條件機率分配剛好就是A 和B，則稱A和B相容。我們透過圖形表示法，提出新的二個離散條件機率分配會相容的充分必要條件。另外，我們證明，二個相容的條件機率分配會有唯一的聯合機率分配的充分必要條件為:所對應的圖形是相連的。我們也討論馬可夫鏈與相容性的關係。 / For two discrete random variables X1 and X2 taking values in {1,…,I} and {1,…,J}, respectively, a putative conditional model for the joint distribution of X1 and X2 consists of two I × J matrices representing the conditional distributions of X1 given X2 and of X2 given X1. We say that two conditional distributions (matrices) A and B are compatible if there exists a joint distribution of X1 and X2 whose two conditional distributions are exactly A and B. We present new versions of necessary and sufficient conditions for compatibility of discrete conditional distributions via a graphical representation. Moreover, we show that there is a unique joint distribution for two given compatible conditional distributions if and only if the corresponding graph is connected. Markov chain characterizations are also presented.

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 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，則如何訂定測量不相容程度的方 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 如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約 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 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。 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 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 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 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 容性問題(一)的部份。 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 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 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 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， 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 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。 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 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 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， 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 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 variables, the issues of the compatibility include: (a) how to determine whether they are compatible? If compatible, how to check the uniqueness of the joint distribution or find all possible joint distributions; (b) if incompatible, how to measure how far they are from compatibility and find the most nearly compatible joint distribution. There are several approaches to solve these problems, such as the ratio matrix method(Arnold and Press, 1989), the IBD matrix method(Song et al., 2010) and the mathematical programming method(Arnold et al., 2002). Inspired by these methods, the thesis develops the eigenvector approach to deal with the compatibility issues. When A and B are compatible, it is observed that the marginal distributions are eigenvectors of AB′ and B′A corresponding to 1, respectively. While checking compatibility by the marginal distributions, the eigenvector approach only checks the marginal distributions which are eigenvectors of AB′ and B′A. It significantly reduces the workload. By using Perron theorem and the concept of the IBD matrix, the part (a) of compatibility issues can be dealt with the eigenvector approach. When A and B are incompatible, a simple way to measure the degree of incompatibility can be derived from the eigenvector approach. In order to compare the most nearly compatible joint distributions given by different measures, the thesis proposes the deviation of the conditional distributions plus the deviation of the marginal distributions as the most nearly compatible joint distribution assessment standard. The eigenvector approach not only derives formula for the most nearly compatible distribution, but also provides better joint distribution than those given by the other measures through the validations under this standard. The part (b) of compatibility issues can also be dealt with the eigenvector approach. Finally, the eigenvector approach is used in solving game problems. In operations research, strategies adopted by both players are assumed to be independent. However, this independent assumption may not be appropriate, since both players can make decisions through the information provided by the payoffs for the game. Let strategies of both players form two conditional probability matrices, then the game problems can be converted into compatibility issues. We can use the concept of generalized compatibility to analyze game solutions and discuss the best strategies for both players in a variety of measurements.