Monotonicity and dummy free property for multi-choice cooperative games

Given a coalition of ann-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, non-decreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.Partially funded by the NSF grant DMS-9024408     

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.     

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.     

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.     

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。     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

Two-step coalition values for multichoice games

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.     

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.     

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。     

The collective value: a new solution for games with coalition structures

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.     

A new basis and the Shapley value

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game in which cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.     

Games with externalities: games in coalition configuration function form

In this paper we introduce a model of cooperative game with externalities which generalizes games in partition function form by allowing players to take part in more than one coalition. We provide an extension of the Shapley value (1953) to these games, which is a generalization of the Myerson value (1977) for games in partition function form. This value is derived by considering an adaptation of an axiomatic characterization of the Myerson value (1977).     

On the relationship between Shapley and Owen values

Owen value is an extension of Shapley value for cooperative games when a particular coalition structure or partition of the set of players is considered in addition. In this paper, we will obtain the Shapley value as an average of Owen values over each set of the same kind of coalition structures, i.e., coalition structures with equal number of sets sharing the same size.     

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。     

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.     

The non-emptiness of the core of a partition function form game

The purpose of this paper is to provide a necessary and sufficient condition for the non-emptiness of the core for partition function form games. We generalize the Bondareva–Shapley condition to partition function form games and present the condition for the non-emptiness of “the pessimistic core”, and “the optimistic core”. The pessimistic (optimistic) core describes the stability in assuming that players in a deviating coalition anticipate the worst (best) reaction from the other players. In addition, we define two other notions of the core based on exogenous partitions. The balanced collections in partition function form games and some economic applications are also provided.     

A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks

Semivalues are solution concepts for cooperative games that assign to each player a weighted sum of his/her marginal contributions to the coalitions, where the weights only depend on the coalition size. The Shapley value and the Banzhaf value are semivalues. Mixed modified semivalues are solutions for cooperative games when we consider a priori coalition blocks in the player set. For all these solutions, a computational procedure is offered in this paper.     

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。     

Evaluation of the Catalonian Parliament, 1980–1984

We analyze here the process of coalition in the Catalonian Parliament. Using Owen's modification of the Shapley value (for games with a priori unions), we show that the actual result was not unreasonable, given the underlying political structure.     

Forming coalitions and the Shapley NTU value

A simple protocol for coalition formation is presented. First, an order of the players is randomly chosen. Then, a coalition grows by sequentially incorporating new members in this order. The protocol is studied in the context of non-transferable utility (NTU) games in characteristic function form. If (weighted) utility transfers are feasible when everybody cooperates, then the expected subgame perfect equilibrium payoff allocation anticipated before any implemented game is the Shapley NTU value.