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.     

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.     

A Shapley function on a class of cooperative fuzzy games

In this paper, we make a study of the Shapley values for cooperative fuzzy games, games with fuzzy coalitions, which admit the representation of rates of players' participation to each coalition. A Shapley function has been introduced by another author as a function which derives the Shapley value from a given pair of a fuzzy game and a fuzzy coalition. However, the previously proposed axioms of the Shapley function can be considered unnatural. Furthermore, the explicit form of the function has been given only on an unnatural class of fuzzy games. We introduce and investigate a more natural class of fuzzy games. Axioms of the Shapley function are renewed and an explicit form of the Shapley function on the natural class is given. We make sure that the obtained Shapley value for a fuzzy game in the natural class has several rational properties. Finally, an illustrative example is given.     

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。     

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。     

模糊联盟合作对策的收益分配研究

针对现实环境中联盟组成的不确定性, 本文研究了具有模糊联盟的合作对策求解问题。提出了模糊联盟合作对策的一种新的分配方式,即平均分摊解,并给出了这种解与模糊联盟合作对策Shapley值一致的充分条件。同时,还提出了模糊联盟合作对策的Shapley值的一个重要性质。最后,结合算例进行了分析论证。     

A simplified expression of the Shapley function for fuzzy game

In this paper, a simplified expression of the Shapley function for games with fuzzy coalition is proposed, which can be regarded as the generalization of Shapley functions defined in some particular games with fuzzy coalition. The simplified expression of the Shapley function is compared with two definitions established by Butnariu, Tsurumi et al. A conclusion is drawn that the simplified expression of the Shapley function is equivalent to Butnariu’s definition when characteristic function is a game with proportional values, and is equivalent to Tsurumi’s definition when characteristic function is a game with Choquet integral forms. Furthermore, from an angle of interaction between two participation levels, the properties of the two games defined by Butnariu and Tsurumi are respectively studied.     

The Shapley function for fuzzy cooperative games with multilinear extension form

In this paper, the definition of the Shapley function for fuzzy cooperative games is given, which is obtained by extending the classical case. The specific expression of the Shapley function for fuzzy cooperative games with multilinear extension form is given, and its existence and uniqueness are discussed. Furthermore, the properties of the Shapley function are researched. Finally, the fuzzy core for this kind of game is defined, and the relationship between the fuzzy core and the Shapley function is shown.     

DEA production games with fuzzy output prices

In DEA production models the technology is assumed to be implicit in the input-output data given by a set of recorded observations. DEA production games assess the benefits to different firms of pooling their resources and sharing their technology. The crisp version of this type of problems has been studied in the literature and methods to obtain stable solutions have been proposed. However, no solution approach exists when there is uncertainty in the unit output prices, a situation that can clearly occur in practice. This paper extends DEA production games to the case of fuzzy unit output prices. In that scenario the total revenue is uncertain and therefore the corresponding allocation among the players is also necessarily uncertain. A core-like solution concept is introduced for these fuzzy games, the Preference Least Core. The computational burden of obtaining allocations of the fuzzy total profit reached through cooperation that belong to the preference least core is high. However, the results presented in the paper permit us to compute the fuzzy total revenue obtained by the grand coalition and a fuzzy allocation in the preference least core by solving a single linear programming model. The application of the proposed approach is illustrated with the analysis of two cooperative production situations originated by data sets from the literature.     

区间合成模糊对策

给出了一种新的合成模糊对策模型——区间合成模糊对策,研究了区间合成模糊对策的区间稳定集、区间核心、区间Shapley值、区间Banzhaf-Coleman势指标以及与子区间模糊对策的关系。区间合成模糊对策作为一种特殊的模糊数合成模糊对策,对于研究其它具有模糊数的模糊合成对策有一定的参考价值。     

模糊合作博弈局中人参与水平间相互作用度量

本文基于经典合作博弈局中人间相互作用现象的有关度量方法,针对具有模糊联盟的合作博弈问题,给出了模糊联盟中各局中人参与水平间相互作用的度量方法,定义了水平间相互独立性概念,建立了反映局中人各参与水平间相互作用平均程度的两指标:Shapley相互作用指标及Banzhaf相互作用指标.同时对于具有k-单调性的模糊合作博弈局中人参与水平间的边缘相互作用的有关性质作了进一步研究,得出了一些新的结论.     

三人模糊联盟合作博弈的最小核心解

研究了联盟是模糊的合作博弈.利用多维线性扩展的方法定义了模糊联盟最小核心解,并推导出三人模糊联盟合作博弈最小核心的计算公式.研究结果发现,多维线性扩展的模糊联盟合作博弈最小核心解是对清晰联盟合作博弈最小核心解的扩展.最后给出三人模糊联盟合作博弈的一个具体事例,证明了此方法的有效性和适用性.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

博弈联盟模糊收益的分配

提出了联盟模糊收益合理分配的一种新方法.首先,在模糊收益α截集上定义了α合理分配集,分析了该分配集与模糊收益Shapley值的关系.接着,给出了模糊收益的α合理Shapley分配函数,对其性质进行了讨论.然后,构造了模糊合理Shapley分配,证明其连续性,得到了联盟模糊收益与模糊合理Shapley分配具有包含关系的结论.     

一类具有限制联盟结构的合作对策的两阶段Shapley值

讨论一类具有限制联盟结构的合作对策,其中局中人通过优先联盟整体参与大联盟的合作,同时优先联盟内部有合取权限结构限制,利用两阶段Shapley值的分配思想并考虑到权限结构对优先联盟内合作的限制,给出了此类合作对策的解。 该解可看做具有联盟结构的合作对策的两阶段Shapley值的推广。 证明了该解满足的公理化条件,并验证了这些条件的独立性。     

基于Choquet积分形式的模糊联盟核心

在具有联盟结构的合作对策中,针对局中人以某种程度参与到合作中的情况,研究了模糊联盟结构的合作对策的收益分配问题。首先,定义了具有模糊联盟结构的合作对策及相关概念。其次,定义了Choquet积分形式的模糊联盟核心,提出了该核心与联盟核心之间的关系,对于强凸联盟对策,证明Choquet积分形式的模糊Owen值属于其所对应的模糊联盟核心。最后通过算例,对该分配模型的可行性进行分析。     

The Shapley value for cooperative games under precedence constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.     

Games on fuzzy communication structures with Choquet players

Myerson (1977) used graph-theoretic ideas to analyze cooperation structures in games. In his model, he considered the players in a cooperative game as vertices of a graph, which undirected edges defined their communication possibilities. He modified the initial games taking into account the graph and he established a fair allocation rule based on applying the Shapley value to the modified game. Now, we consider a fuzzy graph to introduce leveled communications. In this paper players play in a particular cooperative way: they are always interested first in the biggest feasible coalition and second in the greatest level (Choquet players). We propose a modified game for this situation and a rule of the Myerson kind.