摘要
摘要 給定一個有限的合作對局, 會有一個唯一的分配也就是該對局的 Shapley 值. 在這篇論文中, 我們將證明Shapley 值可以完全被一個特殊形式的線性映射所決定. 第一節將介紹 n 人合作對局的一些基本符號, 例如:參賽者, 聯盟, 利益, 真對局, 利益向量和分配. 在第二節, 我們介紹由複函數所組成的複數向量空間中的一個特別的基底, 這些複函數是被定義在一個有限非空集合的冪集合之上. 在這個向量空間中, 這個特別的基底是由簡易函數所構成, 同時也引用向量空間中任意向量相對該基底的座標表示式. 在第三節中, 我們介紹一個集合函數的承載子和冗員, 一些相關的運算性質和一個最重要的例子和結果, 這個結果證明了承載子的任意交集不一定是一個承載子. 在第四節, 我們討論了作用在集合函數上的排列. 在第五節, 我們探討了從複數向量空間中的 n 人合作對局映到 n 維酉空間的映射函數, 這個函數符合Shapley公設－有效性, 對稱性和累加性, 這樣的映射函數是唯一的, 而且擁有一個直觀的意義. 在最後一節, 我們得知Shapley 值可構成真對局的一組分配.
關鍵字
Shapley值並列摘要
Abstract Given a finite cooperative game, there is a unique imputation known as the Shapley value of the game. In this thesis, we show that the Shapley value is completely determined by a special kind of a linear transformation. The basic notations of n-person cooperative game, such as players, coalitions, utilities, proper games, payoff vectors and imputations are introduced in section 1. In section 2, we introduce a nonstandard basis for the complex vector space of all complex-valued functions defined on the power set of a finite nonempty set, the basis consists of the simplest functions in the vector space, we also derive the well-known formula of the coordinates of an arbitrary vector in the vector space relative to the basis. In section 3, the notions of carriers and dummies of a set function are introduced, some elementary properties of them are discussed and the most interest result of this section is an example showing that an intersection of carriers of a set function may not be a carrier. In section 4, the properties of permutations acting on some kind of set functions are discussed. In section 5, we study the map from the complex vector space of all n-person cooperative game into the n-dimensional unitary space satisfying the Shapley axioms－axiom of effectiveness，axiom of symmetry and axiom of aggregation, such a map is unique and its images has a heuristic explanation. In the final section 6,we see that the Shapley value form an imputation of a proper game.