模糊支付合作博弈的广义Shapley函数

基于广义H-差研究了收益是模糊数的合作博弈的广义Shapley函数。首先,对广义H-差的运算做了合理的假设,并以此为基础,给出了区间值合作博弈的广义区间Shapley值的定义和公理体系。然后,根据模糊数与其截集的关系,给出了模糊支付合作博弈的广义Shapley函数的表达式,并用广义有效性、广义哑元性、广义对称性、广义可加性等四条公理刻画了该广义Shapley函数。同时,给出了广义Shapley函数的存在性条件,证明了广义Shapley函数的存在性与唯一性。并且发现,任意的区间值合作博弈的广义区间Shapley值都存在,任意的收益为中心三角模糊数的合作博弈的广义Shapley函数也都存在。另外,本文指出了不能直接利用α—截集博弈的广义区间Shapley值通过集合套理论构造广义Shapley函数。     

区间合作对策在增广系统上的区间Shapley值

给出了区间合作对策在增广系统上的定义,并利用相应的公理体系及区间数运算的性质,构造出区间合作对策在增广系统上的区间Shapley值,论证了当对策为区间凸对策时的区间Shapley值的唯一性,且探讨了该区间Shapley值的一些性质.最后通过算例来说明在此类区间合作对策上所提方法的实用性与有效性.     

具有区间支付的合作对策的区间Shapley值

针对合作对策中支付函数是区间数的情形,利用区间数运算的性质,对Shapley值在经典意义下的三条公理进行拓广,并论证了该形式下的Shapley 函数的唯一形式,并将区间Shapley值方法应用到供应链协调利益分配的实例中.由于支付函数是区间数,本文最终给出的分配的结果也是一个区间数.通过证明可知,由各个联盟对应区间支付范围内的不同实数值所组成的对策是经典合作对策,并且其Shapley值一定包含在区间Shapley值中.     

基于区间与模糊Shapley值的合作收益分配策略

将经典Shapley值三条公理进行拓广,提出具有模糊支付合作对策的Shapley值公理体系。研究一种特殊的模糊支付合作对策,即具有区间支付的合作对策,并且给出了该区间Shapley值形式。根据模糊数和区间数的对应关系,提出模糊支付合作对策的Shapley值,指出该模糊Shapley值是区间支付模糊合作对策的自然模糊延拓。结果表明:对于任意给定置信水平α,若α=1,则模糊Shapley值对应经典合作对策的Shapley值,否则对应具有区间支付合作对策的区间Shapley值。通过模糊数的排序,给出了最优的分配策略。由于对具有模糊支付的合作对策进行比较系统的研究,从而为如何求解局中人参与联盟程度模糊化、支付函数模糊化的合作对策,奠定了一定的基础。     

具有区间联盟值n人对策的Shapley值

本文提出了一类具有区间联盟收益值n人对策的Shapley值.利用区间数运算有关理论,通过建立公理化体系,对具有区间联盟收益值n人对策的Shapley值进行深入研究,证明了这类n人对策Shapley值存在性与唯一性,并给出了此Shapley值的具体表达式及一些性质.最后通过一个算例检验了其有效性与正确性.     

基于风险偏好与满意度的区间值合作对策

研究区间Shapley值一般是以超可加区间值合作对策或凸区间值合作对策为前提,但这限制了区间Shapley值的适用范围。本文以区间数的接受指标及局中人对风险的偏好水平为基础,提出了局中人满意度的概念,并利用满意度对区间值合作对策进行了探讨。通过计算区间值合作对策的局中人与联盟对其区间Shapley值的满意度,来判断区间Shapley值是否被局中人或联盟接受,形成的联盟是否稳定,拓展了区间值合作对策Shapley值的适用范围。同时,得到了当区间值合作对策满足一定条件时满意度的一些性质。     

区间合作博弈Shapley值的矩阵计算方法

在合作博弈中,Shapley单点解按照参与者对联盟的边际贡献率对联盟的收益进行分配.联盟收益具有不确定性,往往不能用精确数值表示,更多学者关注特征函数取值为有限区间的合作博弈(区间合作博弈)的收益分配.文章利用矩阵半张量积,研究区间合作博弈中含有折扣因子的Shapley区间值的矩阵计算.首先利用矩阵的半张量积将合作博弈的特征函数表示为矩阵形式,得到特征函数区间矩阵.然后通过构造区间合作博弈Shapley矩阵,将区间合作博弈的Shapley值(区间)计算转化为矩阵形式.最后利用区间合作博弈Shapley值矩阵公式计算分析航空公司供应链联盟收益的Shapley值.文章给出的区间合作博弈Shapley值的矩阵计算公式形式简洁,为区间合作博弈的研究提供了新的思路.     

区间数的绝对值与区间值函数的极限

研究了区间数的绝对值和区间值函数的极限问题.首先,讨论了区间数的H-差的性质,得到了H-差的两个运算法则;然后,给出了区间数的绝对值概念,并讨论了区间数绝对值的性质;最后,借助区间数的H-差和绝对值的概念,建立了区间值函数极限概念的一种新的表达方式,给出了极限存在的充分必要条件,证明了极限值的唯一性及对加法运算和数乘运算的封闭性.     

区间合作对策核心存在性的进一步讨论

区间合作对策,是研究当联盟收益值为区间数情形时如何进行合理收益分配的数学模型。近年来,其解的存在性与合理性等问题引起了国内外专家的广泛关注。区间核心,是区间合作对策中一个非常稳定的集值解概念。本文首先针对区间核心的存在性进行深入的讨论,通过引入强非均衡,极小强均衡,模单调等概念,从不同角度给出判别区间核心存在性的充分条件。其次,通过引入相关参数,定义了广义区间核心,并给出定理讨论了区间核心与广义区间核心的存在关系。本文的结论将为进一步推动区间合作对策的发展,为解决区间不确定情形下的收益分配问题奠定理论基础。     

基于Choquet积分的直觉模糊联盟合作博弈的Shapley值

本文针对联盟是直觉模糊集的合作博弈Shapley值进行了研究.通过区间Choquet积分得到直觉模糊联盟合作博弈的特征函数为区间数,并研究了该博弈特征函数性质。根据拓展模糊联盟合作博弈Shapley值的计算方法,得到直觉模糊联盟合作博弈Shapley值的计算公式,该计算公式避免了区间数的减法。进一步证明了其满足经典合作博弈Shapley值的公理性。最后通过数值实例说明本文方法的合理性和有效性。     

Necessary and sufficient conditions for interval-valued differentiability

This paper presents necessary and sufficient conditions for generalized Hukuhara differentiability of interval-valued functions and counterexamples of some equivalences previously presented in the literature, for which important results are based on. Moreover, applications of interval generalized Hukuhara differentiability are presented.     

一种资源投入不确定情形下的合作博弈形式及收益分配策略

首先,将经典合作博弈进行扩展,提出了一类模糊联盟合作博弈的通用形式,涵盖常见三种模糊联盟合作博弈,即多线性扩展博弈、比例模糊博弈与Choquet积分模糊博弈.比例模糊博弈、Choquet积分模糊博弈的Shapley值均可以作为一种特定形式下模糊联盟合作博弈的收益分配策略,但是对于多线性扩展博弈的Shapley值一直关注较少,因此利用经典Shapley值构造出多线性扩展博弈的Shapley值,以此作为一种收益分配策略.最后,通过实例分析了常见三类模糊联盟合作博弈的形式及其对应的分配策略,分析收益最大的模糊联盟合作对策形式及最优分配策略,为不确定情形下的合作问题提供了一定的收益分配依据.     

收益模糊合作对策Shapley值的公理化

研究一类收益模糊的合作对策,这类对策联盟的模糊收益值可以用一个闭区间的形式来表示,本文定义了一个拓展的闭区间空间和一些闭区间线性运算算子,证明了这类对策的Shapley值可以用承载性、可替代性和可加性进行了公理化.     

The Shapley value,the Proper Shapley value,and sharing rules for cooperative ventures

In this note, we discuss two solutions for cooperative transferable utility games, namely the Shapley value and the Proper Shapley value. We characterize positive Proper Shapley values by affine invariance and by an axiom that requires proportional allocation of the surplus according to the individual singleton worths in generalized joint venture games. As a counterpart, we show that affine invariance and an axiom that requires equal allocation of the surplus in generalized joint venture games characterize the Shapley value.     

Games with a permission structure - A survey on generalizations and applications

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000