A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.     

An axiomatization of the disjunctive permission value for games with a permission structure

Players that participate in acooperative game with transferable utilities are assumed to be part of apermission structure being a hierarchical organization in which there are players that need permission from other players before they can cooperate. Thus a permission structure limits the possibilities of coalition formation. Various assumptions can be made about how a permission structure affects the cooperation possibilities. In this paper we consider thedisjunctive approach in which it is assumed that each player needs permission from at least one of his predecessors before he can act. We provide an axiomatic characterization of thedisjunctive permission value being theShapley value of a modified game in which we take account of the limited cooperation possibilities.     

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.     

An axiomatization of the Banzhaf value for cooperative games on antimatroids

Cooperative games on antimatroids are cooperative games in which coalition formation is restricted by a combinatorial structure which generalizes permission structures. These games group several well-known families of games which have important applications in economics and politics. The current paper establishes axioms that determine the restricted Banzhaf value for cooperative games on antimatroids. The set of given axioms generalizes the axiomatizations given for the Banzhaf permission values. We also give an axomatization of the restricted Banzhaf value for the smaller class of poset antimatroids. Finally, we apply the above results to auction situations.     

An algorithm for computing the nucleolus of disjunctive non-negative additive games with an acyclic permission structure

A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. In this paper we consider non-negative additive games with an acyclic permission structure. For such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game. The algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries.     

Potentials and reduced games for share functions

A value function for cooperative games with transferable utility assigns to every game a distribution of the payoffs. A value function is efficient if for every such a game it exactly distributes the worth that can be obtained by all players cooperating together. An approach to efficiently allocate the worth of the ‘grand coalition’ is using share functions which assign to every game a vector whose components sum up to one. Every component of this vector is the corresponding players’ share in the total payoff that is to be distributed. In this paper we give characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.     

Games with permission structures: The conjunctive approach

This paper is devoted to the game theoretic analysis of decision situations, in which the players have veto power over the actions undertaken by certain other players. We give a full characterization of the dividends in these games with a permission structure. We find that the collection of these games forms a subspace of the vector space of all games with side payments on a specified player set.Two applications of these results are provided. The first one deals with the projection of additive games on a permission structure. It is shown that the Shapley value of these projected games can be interpreted as an index that measures the power of the players in the permission structure. The second application applies the derived results on games, where the organization structure can be analysed separately from the production capacities of the participating players.     

Banzhaf Measures for Games with Several Levels of Approval in the Input and Output

An axiomatic characterization of ‘a Banzhaf score’ notion is provided for a class of games called (j,k) simple games with a numeric measure associated to the output set, i.e., games with n players, j ordered qualitative alternatives in the input level and k possible ordered quantitative alternatives in the output. Three Banzhaf measures are also introduced which can be used to determine a player's ‘a priori’ value in such a game. We illustrate by means of several real world examples how to compute these measures. Research partially supported by Grant BFM 2003-01314 of the Science and Technology Spanish Ministry and the European Regional Development Fund.     

Weighted Banzhaf values

The family of weighted Banzhaf values for cooperativen-person TU-games is studied. First we introduce the weighted Banzhaf value for an exogenously given vector of positive weights of the players. Then we give an axiomatic characterization of the class of all possible weighted Banzhaf values.     

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.     

Collusion,quarrel, and the Banzhaf value

We provide new, concise characterizations of the Banzhaf value on a fixed player set employing just the standard dummy player property and one of the collusion properties suggested by Haller (Int J Game Theory 23:261–281, 1994) and Malawski (Int J Game Theory 31:47–67, 2002). Within these characterizations, any of the collusion properties can be replaced by additivity and the quarrel property due to the latter author.     

New characterizations of the Owen and Banzhaf–Owen values using the intracoalitional balanced contributions property

In this paper, several characterizations of the Owen and the Banzhaf–Owen values are provided. All the characterizations make use of a property based on the principle of balanced contributions. This property is called the intracoalitional balanced contributions property and was defined by Calvo et al. (Math Soc Sci 31:171–182, 1996).     

Characterizing the Banzhaf and Shapley values assuming limited linearity

This paper presents characterizations of the Banzhaf-Coleman and Shapley-Shubik indices for monotonic simple games. The characterizations are obtained without explicitly requiring that the indices satisfy the linearity assumptionψ (v∧ w) +ψ (v ∨ w) =ψ (v) + ψ (w). The ideas developed are then used to obtain a characterization of the Banzhaf value for the class of alln-person games in characteristic function form.     

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.     

The Owen and Banzhaf–Owen values revisited

In this work, we consider games with coalitional structure. We afford two new parallel axiomatic characterizations for the well-known Owen and Banzhaf–Owen coalitional values. Two properties are common to both characterizations: a property of balanced contributions and a property of neutrality. The results prove that the main difference between these two coalitional values is that the former is efficient, while the latter verifies a property of 2-efficiency.     

拟阵上合作对策的Banzhaf函数

本文结合文[1,2]中关于拟阵上静态结构和动态结构合作对策Shapley函数的描述,探讨了两类拟阵上的Banzhaf函数.通过给出相应的公理体系,论述了两类拟阵上Banzhaf函数的存在性和唯一性,拓展了拟阵上分配指标的研究范围.同时讨论了两类合作对策上Banzhaf函数的有关性质.最后通过算例来说明局中人在此类合作对策中的Banzhaf指标.     

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.     

Amalgamating players,symmetry, and the Banzhaf value

We suggest new characterizations of the Banzhaf value without the symmetry axiom, which reveal that the characterizations by Lehrer (Int J Game Theory 17:89–99, 1988) and Nowak (Int J Game Theory 26:137–141, 1997) as well as most of the characterizations by Casajus (Theory Decis 71:365–372, 2011b) are redundant. Further, we explore symmetry implications of Lehrer’s 2-efficiency axiom.     

具有图结构和联盟结构的Banzhaf值及应用

Banzhaf值是经典可转移效用合作对策中一个著名的分配规则,可以用来评估参与者在对策中的不同作用。本文将Banzhaf值推广到具有联盟结构和图结构的TU-对策中,首先提出并定义了具有联盟结构和图结构的Banzhaf值(简称PL-Banzhaf值),证明了PL-Banzhaf值满足公平性、平衡贡献性和分割分支总贡献性,并给出了该值的两种公理性刻画。其次,讨论了PL-Banzhaf值在跨国天然气管道案例中的应用,并和其他分配规则进行了比较分析。     

Values and potential of games with cooperation structure

Games with cooperation structure are cooperative games with a family offeasible coalitions, that describes which coalitions can negotiate in the game. We study a model ofcooperation structure and the corresponding restricted game, in which the feasible coalitions are those belonging to apartition system. First, we study a recursive procedure for computing the Hart and Mas-Colell potential of these games and we develop the relation between the dividends of Harsanyi in the restricted game and the worths in the original game. The properties ofpartition convex geometries are used to obtain formulas for theShapley andBanzhaf values of the players in the restricted game in terms of the original gamev. Finally, we consider the Owen multilinear extension for the restricted game.The author is grateful to Paul Edelman, Ulrich Faigle and the referees for their comments and suggestions. The proof of Theorem 1 was proposed by the associate editor's referee.