超图结构上合作博弈的赋权Position值

在合作博弈的一般模型中总是假设所有联盟都能形成。不过,在实际中由于受到一些因素的制约,有些联盟是不能形成的。基于此,Myerson提出了具有图通讯结构的合作博弈。Myerson值和Position值是超图博弈上的两个重要分配规则。2005年,Slikker给出了在图博弈上Position值的公理化刻画。但超图博弈上Position值的公理化刻画一直悬而未决。本文通过引入“赋权平衡超边贡献公理”,并结合经典的“分支有效性”,提出了超图博弈上赋权Position值的公理化刻画。作为推论,解决了超图博弈上Position值的公理化刻画问题。     

有效商Myerson值的公理化刻画

在具有图结构的合作对策中，Myerson值(Myerson, 1977)是一个著名的分配规则，它可以由分支有效性和公平性或者平衡贡献性所唯一确定。在实际中，图结构可能并不影响大联盟的形成，只是由于参与者在网络中所处的位置不同，对其讨价还价能力会产生影响。换句话说，图结构会对分配格局产生影响，但对大联盟的形成没有影响。这促使人们开始考虑Myerson值的有效推广问题。文献中已经提出了Myerson的几种有效推广形式。2020年，Li和Shan提出了有效商Myerson值并给出了公理化刻画，它是Myerson值一种新的有效推广形式。本文首先引入了准商盈余公平性这一性质，然后结合有效性和Myerson值黏性给出了有效商Myerson值的新公理化刻画。其次，通过应用案例，将该值和其他值做了比较分析。     

图上合作博弈和图的边密度

1977年, Myerson建立了以图作为合作结构的可转移效用博弈模型(也称图博弈), 并提出了一个分配规则, 也即"Myerson 值", 它推广了著名的Shapley值. 该模型假定每个连通集合(通过边直接或间接内部相连的参与者集合)才能形成可行的合作联盟而取得相应的收益, 而不考虑连通集合的具体结构. 引入图的局部边密度来度量每个连通集合中各成员之间联系的紧密程度, 即以该连通集合的导出子图的边密度来作为他们的收益系数, 并由此定义了具有边密度的Myerson值, 证明了具有边密度的Myerson值可以由"边密度分支有效性"和"公平性"来唯一确定.     

联盟缺额合意性与均分不可分贡献值

文章首先基于联盟盈余合意性(Hu,2019)提出了合作博弈解新的公理,即联盟缺额合意性,并证明了除了不超过2个局中人合作博弈的平凡情形之外,联盟缺额合意性与合作博弈解的有效性互斥.其次,通过对联盟缺额进行平均化引入了平均联盟缺额合意性,进一步结合有效性和可加性实现了均分不可分贡献值的公理化刻画.最后,将相关公理化结果拓展到了权重均分不可分贡献值(Hou等,2019).     

带层次结构效用可转移合作对策的collective值

层次结构是比联盟结构更一般化的结盟限制形式.文章给出了带层次结构效用可转移合作对策collective值的两种定义.第一种类似于带层次结构效用可转移合作对策的ζ值,从多步分配的角度定义了collective值;第二种类似于带图结构效用可转移合作对策的Myerson值,从交流限制的角度定义了collective值.文中证明了这两种定义是等价的,并给出了collective值的一个公理化刻画.研究结果丰富了带层次结构效用可转移合作对策的解概念,可为研究其它带结盟限制效用可转移合作对策的值函数提供借鉴.     

有限合作博弈的Shapley分配

以Myerson关于有限合作的图博弈模型为基础,结合经典合作博弈的相关结论,建立了有限合作博弈的Shapley分配,讨论了分配的相关性质.同时在支付函数满足链递增性的假设下,进一步研究了有限合作关系变化对收益分配的影响,给出了相关的研究结论.     

基于图对策的收益分配

基于具有交流结构的合作对策,即图对策,对平均树解拓展形式的特征进行刻画,提出此解满足可加性公理。进一步地,分析了对于无圈图对策此解是分支有效的。并且当连通分支中两个局中人相关联的边删掉后,此连通分支的收益变化情况可用平均树解表示。这一性质是Shapley值和Myerson值所不具有的。最后,我们给出了模糊联盟图对策中模糊平均树解的可加性和分支有效性。     

直觉模糊联盟合作博弈的非线性规划模型

本文研究联盟是直觉模糊集的合作博弈。首先,给出直觉模糊联盟的定义,并根据Choquet积分的直觉模糊形式,得到直觉模糊联盟合作博弈的区间值特征函数,进一步证明直觉模糊联盟合作博弈的区间值特征函数具有超可加性、凸性、弱超可加性. 其次根据区间数的闵可夫斯基距离、区间数的排序及损失函数的定义,建立直觉模糊联盟合作博弈的非线性规划模型,并对其求解得到最优分配. 最后给出一个具体的事例说明本文所建立的模型的合理性和有效性。     

边赋权图对策Myerson值的和分解

2003年,Gómez等在考虑社会网络中心性度量时,引入了对称对策上Myerson值的和分解概念,本文将这一概念推广到边赋权图对策上,给出了相应于边赋权图对策的组内Myerson值和组间Myerson值。其中边的权表示这条边的两个端点之间的直接通讯容量,组内Myerson值衡量了每个参与者来自它所在联盟的收益,而组间Myerson值评估了参与者作为其他参与者中介所获取的收益。本文侧重分析了边赋权图对策的组内Myerson值和组间Myerson值的权稳定性和广义稳定性, 并给出了这两类值的刻画。     

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。     

具有超图合作结构的赋权Position值

具有超图交流结构的可转移效用合作对策,也称为超图对策,它由一个三元组(N,v,H)所组成,其中(N,H)是一个可转移效用对策(简称TU-对策),而(N,H)是一个超图(超网络)。在超图对策中,除Myerson值(Myerson)外,Position值(Meessen)是另一个重要的分配规则。该模型要求把超图结构中每条超边Shapley的值平均分配给它所包含的点,而不考虑每个点的交流能力或合作水平。本文引入超图结构中点的度值来度量每条超边中每个点的交流能力或合作水平,并结合Haeringer提出用于推广Shapley值的权重系统,并由此定义了具有超图合作结构的赋权Position值。我们证明了具有超图合作结构的赋权Position值可以由“分支有效性”、“冗余超边性”、“超边可分解性”、“拟可加性”、“弱积极性”和“弱能转换”六个性质所唯一确定,并且发现参与者获得的支付随其度值的增加而增加,参与者分摊的成本随其度值的增加而降低。     

The degree value for games with communication structure

A new value concept, called degree value, is proposed by employing the degree game induced by an original game for hypergraph communication situations (including graph communication situations). We provide an axiomatic characterization of the degree value for arbitrary hypergraph communication situations by applying component efficiency and balanced conference contributions, which is a natural extension of balanced link contributions introduced in Slikker (Int J Game Theory 33:505–514, 2005) for graph communication situations. By comparing the degree value with the position value and the Myerson value, it is verified that the degree value is a new allocation rule that differs from both the Myerson value and the position value, and the degree value highlights the important role of the degree of a player in hypergraph communication situations. Particularly, in a uniform hypergraph communication situation, where every conference contains the same number of players, we show that the degree value coincides with the position value.     

The Myerson Value and Superfluous Supports in Union Stable Systems

In this paper, the set of feasible coalitions in a cooperative game is given by a union stable system. Well-known examples of such systems are communication situations and permission structures. Two games associated with a game on a union stable system are the restricted game (on the set of players in the game) and the conference game (on the set of supports of the system). We define two types of superfluous support property through these two games and provide new characterizations for the Myerson value. Finally, we analyze inheritance of properties between the restricted game and the conference game.     

The efficient proportional Myerson values

In this paper we introduce a class of efficient extensions of the Myerson value for games with communication graph structures in which the surplus is allocated in proportion to measures defined on the graph. We show that the efficient proportional Myerson values can be characterized by efficiency, coherence with the Myerson value for connected graphs, and $\alpha$-fairness of surplus. The axiomatization implies a new characterization of the efficient egalitarian Myerson value proposed by van den Brink et al. (2012).     

A linear proportional effort allocation rule

This paper proposes a new class of allocation rules in network games. Like the solution theory in cooperative games of how the Harsanyi dividend of each coalition is distributed among a set of players, this new class of allocation rules focuses on the distribution of the dividend of each network. The dividend of each network is allocated in proportion to some measure of each player’s effort, which is called an effort function. With linearity of the allocation rules, an allocation rule is specified by the effort functions. These types of allocation rules are called linear proportional effort allocation rules. Two famous allocation rules, the Myerson value and the position value, belong to this class of allocation rules. In this study, we provide a unifying approach to define the two aforementioned values. Moreover, we provide an axiomatic analysis of this class of allocation rules, and axiomatize the Myerson value, the position value, and their non-symmetric generalizations in terms of effort functions. We propose a new allocation rule in network games that also belongs to this class of allocation rules.     

The Myerson value for complete coalition structures

In order to describe partial cooperation structures, this paper introduces complete coalition structures as sets of feasible coalitions. A complete coalition structure has a property that, for any coalition, if each pair of players in the coalition belongs to some feasible coalition contained in the coalition then the coalition itself is also feasible. The union stable structures, which constitute the domain of the Myerson value, are a special class of the complete coalition structures. As an allocation rule on complete coalition structures, this paper proposes an extension of the Myerson value for complete coalition structures and provides an axiomatization.