Shapley Value Of Game Research Articles

Complexity of Computing the Shapley Value in Games with Externalities

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value to games with externalities. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets.

A Note on Shapley Ratings in Brain Networks

We consider the problem of computing the influence of a neuronal structure in a brain network. Abraham, Kotter, Krumnack, and Wanke (2006) computed this influence by using the Shapley value of a coalitional game corresponding to a directed network as a rating. Kotter, Reid, Krumnack, Wanke, and Sporns (2007) applied this rating to large-scale brain networks, in particular to the macaque visual cortex and the macaque prefrontal cortex. We introduce an alternative coalitional game that is more intuitive from a game theoretical point of view. We use the Shapley value of this game as an alternative rating to analyze the macaque brain networks and corroborate the findings of Kotter et al. (2007). Moreover, we show how missing information on the existence of certain connections can readily be incorporated into this game and the corresponding Shapley rating.

A Note on a Class of Solutions for Games with Externalities Generalizing the Shapley Value

In this paper we study a family of extensions of the Shapley value for games in partition function form with n players. In particular, we provide a complete characterization for all linear, symmetric, efficient and null solutions in these environments. Finally, we relate our characterization result with other ways to extend the Shapley value in the literature.

A coalitional value for games on convex geometries with a coalition structure

With respect to games on convex geometries with a coalition structure, a coalitional value named the generalized symmetric coalitional Banzhaf value is defined, which can be seen as an extension of the symmetric coalitional Banzhaf value given by Alonso-Meijide and Fiestrs-Janeiro and the Shapley value for games on convex geometries introduced by Bilbao. Based on the established axiomatic system, the existence and uniqueness of the given coalitional value is shown. Meanwhile, a special case is briefly studied.

On the Coalitional Rationality of the Shapley Value and Other Efficient Values of Cooperative TU Games

In the theory of cooperative transferable utilities games, (TU games), the Efficient Values, that is those which show how the win of the grand coalition is shared by the players, may not be a good solution to give a fair outcome to each player. In an earlier work of the author, the Inverse Problem has been stated and explicitely solved for the Shapley Value and for the Least Square Values. In the present paper, for a given vector, which is the Shapley Value of a game, but it is not coalitional rational, that is it does not belong to the Core of the game, we would like to find out a new game with the Shapley Value equal to the a priori given vector and for which this vector is also in the Core of the game. In other words, in the Inverse Set relative to the Shapley Value, we want to find out a new game, for which the Shapley Value is coalitional rational. The results show how such a game may be obtained, and some examples are illustrating the technique. Moreover, it is shown that beside the original game, there are always other games for which the given vector is not in the Core. The similar problem is solved for the Least Square Values.

Marginality Approach to Shapley Value in Games with Externalities

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities. In particular, when externalities occur, a direct translation of Shapley's axioms does not imply a unique value. In this paper we study the marginality approach to this problem, based on the idea of an alpha-parametrized definition of the marginal contribution, where alpha is a vector of weights associated with an agent joining/leaving a coalition. We prove that all values that satisfy the direct translation of Shapley's axioms can be obtained using the marginality approach. Moreover, we show that every such value can be uniquely derived using marginality approach by choosing appropriate weights alpha. Next, we analyze how properties of a value translate to the requirements on the definition of the marginal contribution (i.e. weights alpha). Building upon this analysis, we show that under certain conditions, two other axiomatizations of the Shapley value (i.e., Young's marginality axiomatization and Myerson's axiomatization based on the concept of balanced contributions), translated to games with externalities using the proper definition of the alpha-parametrized marginal contribution, are equivalent to Shapley's axiomatization.

A matrix approach to associated consistency of the Shapley value for games in generalized characteristic function form

The paper is devoted to the Shapley value for cooperative games in generalized characteristic function form in which the players of coalitions are supposed to be ordered. An axiomatization of the generalized Shapley value is presented in terms of three properties, namely continuity, associated consistency, and inessential game property. The proof of the characterization of the Shapley value is based on a matrix representation of so-called associated games. We analyze the dimension of the eigenspaces of this matrix and show that the matrix is diagonalizable.

Games on convex geometries with a coalition structure

In this study, the model of games on convex geometries with a coalition structure is introduced, where the players can participate in different unions. The Shapley value for games on convex geometries with a coalition structure is researched, which can be seen as an extension of the Shapley value for games on convex geometries and that for the traditional games, respectively. By establishing the associated axiomatic systems, the existence and uniqueness of the given Shapley value is proved. Meantime, the core for this kind of games is defined, and an equivalent form is studied. When the given game is convex, some properties are researched.

PROJECT MANAGEMENT GAMES

This paper studies situations in which companies can cooperate in order to decrease the earliest completion time of a project that consists of several tasks. This is beneficial for the client who wants the project to be completed as early as possible. The client is willing to pay more for an earlier completion time. The total payoff must be allocated among the companies that cooperate. Cooperative game theory is used to model this situation. Conditions for the core of the game to be nonempty are derived. We study a class of project management games for which necessary and sufficient conditions for the nonemptiness of the core can be derived. We will show that a subset of the set of balanced project management games can be partitioned into a class of 1-convex games and a class of big boss games. Expressions for the extreme points of the core, the τ-value, the nucleolus, and the Shapley-value of games in these two classes are derived.

A Museum Cost Sharing Problem

Ginsburgh and Zang [2] consider a revenue sharing problem for the museum pass program, in which several museums jointly offer museum passes that allow visitors an unlimited access to participating museums in a certain period of time. We consider a cost sharing problem that can be regarded as the dual problem of the above revenue sharing problem. We assume that all museums are public goods and have various (e.g., ser-vice) costs. These costs must be shared by museum visitors. We propose a cost sharing method and provide an axiomatic characterization of the method. We then define a game for the problem and show that the cost sharing method is the Shapley value of the game. We also provide a comparative statics analysis for both the Shapley value of the museum pass game and the Shapley value for the cost sharing game when the number of museums and/or the number of visitors change.

The Shapley Value for Games with Restricted Cooperation

The traditional assumption in cooperative game theory is that every coalition is feasible and can form to attain its payoff. However, in many real life situations not every group of players has the opportunity to cooperate and to collect their own payoff. We say that we deal with cooperative games with restricted cooperation when not all coalitions can form. In this paper we will deal with generalizations of the Shapley value for games with restricted cooperation. Three solutions for games with restricted cooperation will be considered. One of them (the Myerson value) is well known. Two others are based on the same principle: to construct some restricted game and to use the Shapley value for this game. We consider the class of all solutions which can be constructed in an analogous way. We will show that this class coincides with the class of all solutions with balanced contribution property. Also each solution from this class can be described by either a potential function or a consistency property.

Axiomatizations of the Shapley value for games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed another model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.

RANKING BY RELATIONAL POWER BASED ON DIGRAPHS

In this paper we examine the ranking of objects whose relative merits are given by a directed graph. We consider several measures and show their rationality through axiomatization as well as showing the relationship with the Shapley value of games whose characteristic function is derived from the directed graph. We also give some numerical examples and report the experience of the application to the best paper selection problem.

Axiomatizations of Two Types of Shapley Values for Games on Union Closed Systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson rule for conference structures.

Axiomatization of Shapley Values of Fagle and Kern Type on Set Systems

We propose axiomatizing a generalized Shapley value of games for potential application to games on set systems satisfying the condition of normality. This encompasses both the original Shapley value and Faigle and Kern's Shapley value, which is generalized for a cooperative game defined on a subcoalition.

Potential approach and characterizations of a Shapley value in multi-choice games

The main focus of this paper is on the restricted Shapley value for multi-choice games introduced by Derks and Peters [Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game Theory 21, 351–360] and studied by Klijn et al. [Klijn, F., Slikker, M., Zazuelo, J., 1999. Characterizations of a multi-choice value. International Journal of Game Theory 28, 521–532]. We adopt several characterizations from TU game theory and reinterpret them in the framework of multi-choice games. We generalize the potential approach and show that this solution can be formulated as the vector of marginal contributions of a potential function. Also, we characterize the family of all solutions for multi-choice games that admit a potential. Further, a consistency result is reported.

On the Semivalues and the Least Square Values Average Per Capita Formulas and Relationships

In this paper, it is shown that both the Semivalues and the Least Square Values of cooperative transferable utilities games can be expressed in terms of n2 averages of values of the characteristic function of the game, by means of what we call the Average per capita formulas. Moreover, like the case of the Shapley value earlier considered, the terms of the formulas can be computed in parallel, and an algorithm is derived. From these results, it follows that each of the two values mentioned above are Shapley values of games easily obtained from the given game, and this fact gives another computational opportunity, as soon as the computation of the Shapley value is efficiently done.

The Probabilistic Representative Values

In this paper we define a new family of solutions for the class of cooperative games with transferable utility, in which the set of players exhibits a structure of a priori unions.This family is deeply connected with the Shapley value for games with transferable utility but, moreover, we assume a solidarity strong connection among all the components of each union.As a consequence of this, they are disposed to delegate one coalition of members of the union to negotiate with the other unions, and, therefore, each union will have a representative coalition.Furthermore, three interesting solutions that belong to this family of values are studied, as well as the non cooperative selection of the best representative coalition for each union.

"Counting'' power indices for games with a priori unions

The classical Owen construction of the Shapley value for games with a priori unions is adapted to extend the class of "counting" power indices - i.e., those computed by counting appropriately weighted contributions of players to winning coalitions - to simple games with a priori unions. This class contains most well-known indices, including Ban­ zhaf, Johnston, Holler and Deegan-Packel indices. The Shapley-Shubik index for simple games with a priori unions coincides with the (restriction of) Shapley value for such games obtained by Owen's method, but for all other indices obtained by normalization of probabilistic values our con­ struction leads to indices different from those determined by Owen values.

A Rough Set Approach to Estimating the Game Value and the Shapley Value from Data

A value of a game v is a function which to each coalition S assigns the value v(S) of this coalition, meaning the expected pay–off for players in that coalition. A classical approach of von Neumann and Morgenstern [6] had set some formal requirements on v which contemporary theories of value adhere to. A Shapley value of the game with a value v [14] is a functional Φ giving for each player p the value Φp(v) estimating the expected pay-off of the player p in the game. Game as well as conflict theory have been given recently much attention on the part of rough and fuzzy set communities [11,8,1,4,7,2]. In particular, problems of plausible strategies [1] in conflicts as well as problems related to Shapley's value [3,2] have been addressed.We confront here the problem of estimating a value as well as Shapley's value of a game from a partial data about the game. We apply to this end the rough set ideas of approximations, defining the lower and the upper value of the game and, respectively, the lower and upper Shapley value. We also define a notion of an exact coalition, on which both values coincide giving the true value of the game; we investigate the structure of the family of exact sets showing its closeness on complements, disjoint sums, and intersections of coalitions covering the set of players. This work sets open a new area of rough set applications in mining constructs from data. The construct mined in this case are values as well as Shapley values of games.