拟阵上模糊合作对策的Banzhaf函数

首先通过对清晰拟阵定义的拓展,给出了模糊拟阵的概念。通过定义具有多线性扩展形式的模糊合作对策在静态结构和动态结构拟阵上B anzhaf函数的公理体系,分别探讨了此类模糊合作对策在这两种拟阵上关于B anzhaf函数的存在性和唯一性。同时,通过定义具有Choquet积分形式模糊合作对策在静态结构和动态结构拟阵上B anzhaf函数的公理体系,分别探讨了此类模糊合作对策在这两种拟阵上关于B anzhaf函数的存在性和唯一性。     

拟阵限制下合作对策解的传递性

Vincent Feltkamp研究了Shapley解和Banzhaf解的公理性．Bilbao等人又对拟阵限制下的Shapley解的性质进行了讨论．本文在此基础上主要研究了拟阵限制下的合作对策Shapley解，并利用传递性、交换性、概率有效性和P-哑元性等四条公理证明了拟阵限制下合作对策Shapley解的唯一性．进而证明了拟阵限制条件下简单对策Shapley解的唯一性．最后给出了拟阵限制下合作对策的Banzhaf解的唯一性定理．     

拟阵上动态结构合作对策的单调解

论文主要介绍了拟阵上动态结构合作对策单调解的结构,并利用强单调性、交换性和动态有效性等三条公理刻画了此单调解的唯一性.同时给出了拟阵上动态结构合作对策核心的定义,确定了它的结构.最后讨论了拟阵上动态结构合作对策Shapley值与其核心的关系.     

具有图结构和联盟结构的Banzhaf值及应用

Banzhaf值是经典可转移效用合作对策中一个著名的分配规则,可以用来评估参与者在对策中的不同作用。本文将Banzhaf值推广到具有联盟结构和图结构的TU-对策中,首先提出并定义了具有联盟结构和图结构的Banzhaf值(简称PL-Banzhaf值),证明了PL-Banzhaf值满足公平性、平衡贡献性和分割分支总贡献性,并给出了该值的两种公理性刻画。其次,讨论了PL-Banzhaf值在跨国天然气管道案例中的应用,并和其他分配规则进行了比较分析。     

拟阵上合作对策的单调解

本文主要介绍了拟阵上的合作对策Shapley解的结构,并利用强单调性、交换性、概率有效性等三条公理刻画了拟阵上合作对策Shapley解的唯-性.同时讨论了本文的三条公理与Bilbao等人的四条 公理的等价性.最后给出拟阵上合作对策核心的定义及其结构.     

格上合作对策的Banzhaf值

本文介绍了格上的合作对策,并给出了格上合作对策的Banzhaf解,同时利用线性性,哑元性,单调性,对称性,不变性和2-有效性等六条公理完成了对Banzhaf值的唯一性刻画.在证明唯一性的过程中,利用了一个同构变换,将格上多选择合作对策映射到经典合作对策来进行研究.     

集合对策的定量边缘解

本文提出了集合对策的两类定量边缘解,并给出了两类解的公理化特征:有效性、对称性、哑元性、Banzhaf总和性和传递性．这两类解分别与TU-对策的Banzhaf权力指数和Shapley-Shubik权力指数类似．同时,本文将Shapley解与Banzzhaf解扩展到k-维欧氏空间．     

具有多联盟结构的扩展型部分合作对策

通过定义新的合作函数,得到具有多联盟结构的扩展型部分合作对策,并运用逆推归纳法建立部分合作对策解的概念,构造出相应的最优路径. 模型克服了经典合作对策模型中对策树上任意结点处只能形成简单联盟结构的局限性.     

Banzhaf权力指标的特性

Banzhaf权力指标主要用来刻画政治选举中权力的分配问题.本文研究了Banzhaf权力指标的四条性质,它们分别为2人有效性,哑元性,等价性和边缘贡献性.并利用这四条性质刻画了Banzhaf权力指标的唯一性.最后通过一个实例说明了Banzhaf权力指标的应用.     

有向图对策下的Banzhaf值及其应用

Banzhaf值是经典可转移效用合作对策中重要的分配规则之一，它假设任何有限参与者间均能进行合作形成可行联盟。2006年，Alonso-Meijide和Fiestras-Janeiro考虑无向网络，定义了图对策下的Banzhaf值，以此反映合作网络对参与者间合作以及分配结果的影响。本文则在此基础上，考虑合作网络的方向性，将Banzhaf值进一步推广到有向图对策中，提出了新的分配规则——有向Banzhaf值。首先，本文证明了有向Banzhaf值满足准隔离性、收缩性、公平性、强分支可分解性以及强分支总贡献性。其次，证明了有向Banzhaf值可由公平性、准隔离性以及收缩性唯一刻画，也可由公平性结合强分支总贡献性唯一刻画。最后，以湿地水循环系统为例，对有向Banzhaf值和其他值进行了比较分析，讨论了有向Banzhaf值的应用价值。     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

The fuzzy core and Shapley function for dynamic fuzzy games on matroids

In this paper, the generalized forms of the fuzzy core and the Shapley function for dynamic fuzzy games on matroids are given. An equivalent form of the fuzzy core is researched. In order to better understand the fuzzy core and the Shapley function for dynamic fuzzy games on matroids, we pay more attention to study three kinds of dynamic fuzzy games on matroids, which are named as fuzzy games with multilinear extension form, with proportional value and with Choquet integral form, respectively. Meantime, the relationship between the fuzzy core and the Shapley function for dynamic fuzzy games on matroids is researched, which coincides with the crisp case.     

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ³ and that of Motzkin on ≤12 lines in ℝ², extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.     

A general model for matroids and the greedy algorithm

We present a general model for set systems to be independence families with respect to set families which determine classes of proper weight functions on a ground set. Within this model, matroids arise from a natural subclass and can be characterized by the optimality of the greedy algorithm. This model includes and extends many of the models for generalized matroid-type greedy algorithms proposed in the literature and, in particular, integral polymatroids. We discuss the relationship between these general matroids and classical matroids and provide a Dilworth embedding that allows us to represent matroids with underlying partial order structures within classical matroids. Whether a similar representation is possible for matroids on convex geometries is an open question. S. Fujishige’s research was supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).     

Directed submodularity,ditroids and directed submodular flows

Set relations and operations such as inclusion, union and intersection are generalized to directed subsets whose elements are distinguished between forward and backward elements. The concepts of submodular functions, matroids and polymatroidal network flows are extended to the concepts of directed submodular functions, ditroids and directed submodular flows on directed subsets. Two unrelated matroids (submodular functions) can be embedded in one ditroid (directed submodular function). Total dual integrality is preserved in these generalizations and proved for very general set-function class-directed odd submodular functions.This work was partially supported by Chinese National Natural Science Fund.     

关于模糊拟阵的独立子集套和独立集函数

本文的研究方法主要是将模糊拟阵问题转化为普通拟阵问题来研究的方法。本文首先建立独立子集套概念,并使用这个概念和独立集函数概念构建了闭模糊拟阵的充要条件和模糊独立集的充要条件;然后,本文仔细分析了模糊基的性质,找到了一个使用独立子集套和独立集函数来描述的模糊基的充要条件;最后,利用模糊基的这个充要条件提出并证明了闭正规模糊拟阵的充要条件。     

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.     

An axiomatic approach to the concept of interaction among players in cooperative games

An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of pseudo-Boolean functions. Received: October 1997/revised version: October 1998