Interaction indices for games on combinatorial structures with forbidden coalitions

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints.     

The Myerson value for directed graph games

A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraph-restricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.     

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.     

The Shapley value for the probability game

The main goal of this paper is to introduce the probability game. On one hand, we analyze the Shapley value by providing an axiomatic characterization. We propose the so-called independent fairness property, meaning that for any two players, the player with larger individual value gets a larger portion of the total benefit. On the other, we use the Shapley value for studying the profitability of merging two agents.     

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

This study provides a unified axiomatic characterization method of one-point solutions for cooperative games with transferable utilities. Any one-point solution that satisfies efficiency, the balanced cycle contributions property (BCC), and the axioms related to invariance under a player deletion is characterized as a corollary of our general result. BCC is a weaker requirement than the well-known balanced contributions property. Any one-point solution that is both symmetric and linear satisfies BCC. The invariance axioms necessitate that the deletion of a specific player from games does not affect the other players’ payoffs, and this deletion is different with respect to solutions. As corollaries of the above characterization result, we are able to characterize the well-known one-point solutions, the Shapley, egalitarian, and solidarity values, in a unified manner. We also studied characterizations of an inefficient one-point solution, the Banzhaf value that is a well-known alternative to the Shapley value.     

A value for partially defined cooperative games

The Shapley value provides a method, which satisfies certain desirable axioms, of allocating benefits to the players of a cooperative game. When there aren players andn is large, the Shapley value requires a large amount of accounting because the number of coalitions grows exponentially withn. This paper proposes a modified value that shares some of the axiomatic properties of the Shapley value yet allows the consideration of games that are defined only for certain coalitions. Two different axiom systems are shown to determine the same modified value uniquely.     

On the symmetric and weighted shapley values

We present new axiomatic characterizations of the symmetric Shapley value and of weighted Shapley values for transferable utility coalitional form games without imposing the axiom ofadditivity (Shapley [1953a,b]). Our main condition iscoalitional strategic equivalence, introduced by Chun [1989]. We show thatcoalitional strategic equivalence, together withefficiency, andsymmetry, characterizes the symmetric Shapley value, and this axiom, together withefficiency, positivity, homogeneity, andpartnership, characterizes weighted Shapley values.     

Egalitarian allocation and players of certain type

In cooperative game theory the Shapley value is different from the egalitarian value, the latter of which allocates payoffs equally. The null player property and the nullifying player property assign zero payoff to each null player and each nullifying player, respectively. It is known that if the null player property for characterizing the Shapley value is replaced by the nullifying player property, then the egalitarian value is determined uniquely. We propose several properties to replace the nullifying player property to characterize the egalitarian value. Roughly speaking, the results in this note hint that equal division for players of certain types may lead to the egalitarian allocation.     

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).     

Random order coalition structure values

Recently, attention has been focused on generalizations of the Shapley value obtained by relaxing the symmetry postulate. Shapley defined the class of weighted values and these have been characterized by Kalai and Samet. Random order values, treated by Weber, provide the most general approach to values without symmetry. This paper extends the random order idea to games with coalition structures. The symmetric CS value was defined by Owen; axiomatic characterizations have been given by Owen and Hart and Kurz. Levy and McLean extended their work and analyzed various classes of weighted CS values. The random order CS values of this paper include all the CS values described above as special cases.     

A new axiomatization of the Shapley–solidarity value for games with a coalition structure

In this paper, we propose a new kind of players as a compromise between the null player and the A-null player. It turns out that the axiom requiring this kind of players to get zero-payoff together with the well-known axioms of efficiency, additivity, coalitional symmetry, and intra-coalitional symmetry characterize the Shapley–solidarity value. This way, the difference between the Shapely–solidarity value and the Owen value is pinpointed to just one axiom.     

最小成本生成树对策上Shapley值的新刻画及其应用

2002年,Kar利用有效性、无交叉补贴性、群独立性和等处理性四个公理对最小成本生成树对策上的Shapley值进行了刻画。本文提出了“群有效性”这一公理,利用这一公理和“等处理性”两个公理,给出了最小成本生成树对策上Shapley值的一种新的公理化刻画。最后,运用最小成本生成树对策的Shapley值,对网络服务的费用分摊问题进行了分析。     

The Shapley value for bicooperative games

The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000) , we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.     

基于平均树值的无圈图博弈有效解

本文对无圈图博弈进行了研究,考虑了大联盟收益不小于各分支收益之和的情况。通过引入剩余公平分配性质,也就是任意两个分支联盟的平均支付变化相等,给出了一个基于平均树值的无圈图博弈有效解。同时,结合有效性和分支公平性对该有效解进行了刻画。特别地,若无圈图博弈满足超可加性时,证明了该有效解一定是核中的元素,说明此时的解是稳定的。最后,通过一案例分析了该有效解的特点,即越大的分支分得的剩余越多,并且关键参与者,也就是具有较大度的参与者可获得相对多的支付。     

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.     

Characterizations of a multi-choice value

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. We study the extended Shapley value as proposed by Derks and Peters (1993). Van den Nouweland (1993) provided a characterization that is an extension of Young's (1985) characterization of the Shapley value. Here we provide several other characterizations, one of which is the analogue of Shapley's (1953) original characterization. The three other characterizations are inspired by Myerson's (1980) characterization of the Shapley value using balanced contributions. Received: November 1997/final version: February 1999     

A value for games withn players andr alternatives

We study value theory for a class of games called games withn players andr alternatives. In these games, each of then players must choose one and only one of ther alternatives. A linear, efficient value is obtained using three characterizations, two of which are axiomatic. This value yields an a priori evaluation for each player relative to each alternative.     

Harsanyi power solutions for graph-restricted games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.     

Alternative axiomatic characterizations of the Shapley and Banzhaf values

In a paper in 1975, Dubey characterized the Shapley-Shubik index axiomatically on the class of monotonic simple games. In 1979, Dubey and Shapley characterized the Banzhaf index in a similar way. This paper extends these characterizations to axiomatic characterizations of the Shapley and Banzhaf values on the class of control games, on the class of simple games and on the class of all transferable utility games. In particular, it is shown that the additivity axiom which is usually used to characterize these values on the class of all transferable utility games can be weakened without changing the result.This research is sponsored by the Foundation for the Promotion of Research in Economic Sciences, which is part of the Dutch Organization for Scientific Research (NWO).     

On bargaining position descriptions in non-transferable utility games

We present a generalization to the Harsanyi solution for non-transferable utility (NTU) games based on non-symmetry among the players. Our notion of non-symmetry is presented by a configuration of weights which correspond to players' relative bargaining power in various coalitions. We show not only that our solution (i.e., the bargaining position solution) generalizes the Harsanyi solution, (and thus also the Shapley value), but also that almost all the non-symmetric generalizations of the Shapley value for transferable utility games known in the literature are in fact bargaining position solutions. We also show that the non-symmetric Nash solution for the bargaining problem is also a special case of our general solution. We use our general representation of non-symmetry to make a detailed comparison of all the recent extensions of the Shapley value using both a direct and an axiomatic approach.