An axiomatization of the Shapley value using a fairness property

In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games. Revised August 2001     

Constrained core solutions for totally positive games with ordered players

In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.     

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 new look at the role of players’ weights in the weighted Shapley value

everal new families of semivalues for weighted n-person transferable utility games are axiomatically constructed and discussed under increasing collections of axioms, where the weighted Shapley value arises as the resulting one member family. A more general approach to such weighted games defined in the form of two components, a weight vector λ and a classical TU-game v, is provided. The proposed axiomatizations are done both in terms of λ and v. Several new axioms related to the weight vector λ are discussed, including the so-called “amalgamating payoffs” axiom, which characterizes the value of a weighted game in terms of another game with a smaller number of players. They allow for a new look at the role of players’ weights in the context of the weighted Shapley value for the model of weighted games, giving new properties of it. Besides, another simple formula for the weighted Shapley value is found and examples illustrating some surprising behavior of it in the context of players’ weights are given. The paper contains a wide discussion of the results obtained.     

全对策的边际贡献值

全对策是定义在局中人集合的所有分划集上的一类特殊合作对策.本文在效用可转移情形下研究全对策的"值"问题.定义了全对策的边际贡献值,得出全对策的Shapley值,以及具有某些性质的值是边际贡献值,并给出两种边际贡献值的具体表达式,及其一些性质.     

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.     

Cooperative games of choosing partners and forming coalitions in the marketplace

Two games of interacting between a coalition of players in a marketplace and the residual players acting there are discussed, along with two approaches to fair imputation of gains of coalitions in cooperative games that are based on the concepts of the Shapley vector and core of a cooperative game. In the first game, which is an antagonistic one, the residual players try to minimize the coalition's gain, whereas in the second game, which is a noncooperative one, they try to maximize their own gain as a coalition. A meaningful interpretation of possible relations between gains and Nash equilibrium strategies in both games considered as those played between a coalition of firms and its surrounding in a particular marketplace in the framework of two classes of n-person games is presented. A particular class of games of choosing partners and forming coalitions in which models of firms operating in the marketplace are those with linear constraints and utility functions being sums of linear and bilinear functions of two corresponding vector arguments is analyzed, and a set of maximin problems on polyhedral sets of connected strategies which the problem of choosing a coalition for a particular firm is reducible to are formulated based on the firm models of the considered kind.     

Nonsymmetric values under Hart-Mass-Colell consistency

Let f be a single valued solution for cooperative TU games that satisfies inessential game property, efficiency, Hart Mas-Colell consistency and for two person games is strictly monotonic and individually unbounded. Then there exists a family of strictly increasing functions associated with players that completely determines f. For two person games, both players have equal differences between their functions at the solution point and at the values of characteristic function of their singletons. This solution for two person games is uniquely extended to n person games due to consistency and efficiency. The extension uses the potential with respect to the family of functions and generalizes potentials introduced by Hart and Mas Colell [6]. The weighted Shapley values, the proportional value described by Ortmann [11], and new values generated by power functions are among these solutions. The author is grateful to anonymous referee and Associate Editor for their comments and suggestions.     

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.     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000     

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.     

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。     

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 consistent Shapley value for hyperplane games

A new value is defined for n-person hyperplane games, i.e., non-sidepayment cooperative games, such that for each coalition, the Pareto optimal set is linear. This is a generalization of the Shapley value for side-payment games. It is shown that this value is consistent in the sense that the payoff in a given game is related to payoffs in reduced games (obtained by excluding some players) in such a way that corrections demanded by coalitions of a fixed size are cancelled out. Moreover, this is the only consistent value which satisfies Pareto optimality (for the grand coalition), symmetry and covariancy with respect to utility changes of scales. It can be reached by players who start from an arbitrary Pareto optimal payoff vector and make successive adjustments.     

Values on regular games under Kirchhoff’s laws

The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff’s laws in the circuit associated to the Hasse diagram of feasible coalitions.     

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.     

On the Convenience to Form Coalitions or Partnerships in Simple Games

A partnership in a cooperative game is a coalition that possesses an internal structure and, simultaneously, behaves as an individual member. Forming partnerships leads to a modification of the original game which differs from the quotient game that arises when one or more coalitions are actually formed. In this paper, the Shapley value is used to discuss the convenience to form either coalitions or partnerships. To this end, the difference between the additive Shapley value of the partnership in the partnership game and the Shapley alliance value of the coalition, and also between the corresponding value of the internal and external players, are analysed. Simple games are especially considered. Research partially supported by Grant BFM 2003-01314 of the Science and Technology Spanish Ministry and the European Regional Development Fund.     

An axiomatic characterization of the Owen–Shapley spatial power index

We present an axiomatic characterization of the Owen–Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent.     

Deposit games with reinvestment

In a deposit game coalitions are formed by players combining their capital. The proceeds of their investments then have to be divided among those players. The current model extends earlier work on capital deposits by allowing reinvestment of returns. Two specific subclasses of deposit games are introduced. These subclasses provide insight in two extreme cases. It is seen that each term dependent deposit game possesses a core element. Capital dependent deposit games are also shown to have a core element and even a population monotonic allocation scheme if the revenue function exhibits increasing returns to scale. Furthermore, it is shown that all superadditive games are deposit games if one allows for debt.     

From hierarchies to levels: new solutions for games with hierarchical structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many of these problems, players are organized according to either a hierarchical structure or a levels structure that restrict the players’ possibilities to cooperate. In this paper, we propose three new solutions for games with hierarchical structure and characterize them by properties that relate a player’s payoff to the payoffs of other players located in specific positions in the hierarchical structure relative to that player. To define each solution, we consider a certain mapping that transforms the hierarchical structure into a levels structure, and then we apply the standard generalization of the Shapley value to the class of games with levels structure. Such transformation mappings are studied by means of properties that relate a player’s position in both types of structure.