Multilinear extensions and values for multichoice games

We define multilinear extensions for multichoice games and relate them to probabilistic values and semivalues. We apply multilinear extensions to show that the Banzhaf value for a compound multichoice game is not the product of the Banzhaf values of the component games, in contrast to the behavior in simple games. Following Owen (Manag Sci 18:64–79, 1972), we integrate the multilinear extension over a simplex to construct a version of the Shapley value for multichoice games. We compare this new Shapley value to other extensions of the Shapley value to multichoice games. We also show how the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of a multichoice game is equal to the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of an appropriately defined TU decomposition game. Finally, we explain how semivalues, probabilistic values, the Banzhaf value, and this Shapley value may be viewed as the probability that a player makes a difference to the outcome of a simple multichoice game.     

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.     

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.     

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.     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

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.     

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 value for multichoice games

A multichoice game is a generalization of a cooperative TU game in which each player has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and related solution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41, 289–311]. We show that this solution applied to the discrete cost sharing model coincides with the Aumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.     

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.     

Compensations in the Shapley value and the compensation solutions for graph games

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give a representation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph in order to construct new allocation rules called the compensation solutions. Firstly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees (see Demange, J Political Econ 112:754–778, 2004) instead of orderings of the players by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively. Secondly, we consider cooperative games with a forest (cycle-free graph) and all its rooted spanning trees. The compensation solution is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component in the communication graph.     

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).     

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     

The Egalitarian Solution for Multichoice Games

In a multichoice game a coalition is characterized by the level at which each player is acting, and to each coalition a real number is assigned. A multichoice solution assigns, for each multichoice game, a numerical value to each possible activity level of each player, intended to measure the contribution of each such level to reaching the grand coalition in which each player is active at the maximal level. The paper focuses on the egalitarian multichoice solution, characterized by the properties of Efficiency, Zero Contribution, Additivity, Anonymity, and Level Symmetry. The egalitarian solution is also shown to satisfy the property of marginalism: it measures the effect of lowering, ceteris paribus, a certain activity level by one. The solution is compared to a multichoice solution studied in Klijn, Slikker, and Zarzuelo (1999). Finally, it is discussed how the formalism of this paper can be applied to the different framework of multi-attribute utilities.     

An axiomatization of probabilistic Owen value for games with coalition structure

In the framework of games with coalition structure, we introduce probabilistic Owen value which is an extension of the Owen value and probabilistic Shapley value by considering the situation that not all priori unions are able to cooperate with others. Then we use five axioms of probabilistic efficiency, symmetric within coalitions, symmetric across coalitions applying to unanimity games, strong monotone property and linearity to axiomatize the value.     

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.     

Pyramidal values

We propose and analyze a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary, and the rest of his marginal contribution to the just formed coalition is distributed among the incumbent players. We relate the pyramidal-type sharing scheme we propose with other sharing schemes, and we also obtain some known values by means of this kind of pyramidal procedures. In particular, we show that the Shapley value can be obtained by means of an interesting pyramidal procedure that distributes nonzero dividends among the incumbents. As a result, we obtain an alternative formulation of the Shapley value based on a measure of complementarity between two players. Finally, we introduce the family of proportional pyramidal values, in which an incumbent receives a dividend in proportion to his initial investment, measured by means of his marginal contribution.     

Weighted values and the core in NTU games

Monderer et al. (Int J Game Theory 21(1):27–39, 1992) proved that the core is included in the set of the weighted Shapley values in TU games. The purpose of this paper is to extend this result to NTU games. We first show that the core is included in the closure of the positively weighted egalitarian solutions introduced by Kalai and Samet (Econometrica 53(2):307–327, 1985). Next, we show that the weighted version of the Shapley NTU value by Shapley (La Decision, aggregation et dynamique des ordres de preference, Editions du Centre National de la Recherche Scientifique, Paris, pp 251–263, 1969) does not always include the core. These results indicate that, in view of the relationship to the core, the egalitarian solution is a more desirable extension of the weighted Shapley value to NTU games. As a byproduct of our approach, we also clarify the relationship between the core and marginal contributions in NTU games. We show that, if the attainable payoff for the grand coalition is represented as a closed-half space, then any element of the core is attainable as the expected value of marginal contributions.     

The interval Shapley value: an axiomatization

The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.     

The Shapley Valuation Function for Strategic Games in which Players Cooperate

In this note we use the Shapley value to define a valuation function. A valuation function associates with every non-empty coalition of players in a strategic game a vector of payoffs for the members of the coalition that provides these players’ valuations of cooperating in the coalition. The Shapley valuation function is defined using the lower-value based method to associate coalitional games with strategic games that was introduced in Carpente et al. (2005). We discuss axiomatic characterizations of the Shapley valuation function.     

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.