The Shapley value for games on matroids: The dynamic model (original) (raw)

Axiomatizations of the Shapley value for cooperative games on antimatroids

Mathematical Methods of Operations Research (ZOR), 2003

Games on antimatroids are cooperative games restricted by a combinatorial structure which generalize the permission structure. So, cooperative games on antimatroids group several well-known families of games which have important applications in economic and politic. Therefore, the study of the rectricted games by antimatroids allows to unify criteria of various lines of research. The current paper establishes axioms that determine the restricted Shapley value on antimatroids by conditions on the cooperative game v and the structure determined by the antimatroid. This axiomatization generalizes the axiomatizations of both the conjunctive and disjunctive permission value for games with a permission structure. We also provide an axiomatization of the Shapley value restricted to the smaller class of poset antimatroids.

Axiomatizations of the Shapley value for games on augmenting systems

European Journal of Operational Research, 2009

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.

Games on lattices, multichoice games and the Shapley value: a new approach

Mathematical Methods of Operations Research, 2007

Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.

Theτ-value for games on matroids

Top, 2002

In the classical model of games with transferable utility one assumes that each subgroup of players can form and cooperate to obtain its value. However, we can think that in some situations this assumption is not realistic, that is, not all coalitions are feasible. This suggests that it is necessary to raise the whole question of generalizing the concept of transferable utility game, and therefore to introduce new solution concepts. In this paper we define games on matroids and extend the τ-value as a compromise value for these games.

Axiomatizations of two types of Shapley values for games on union closed systems

Economic Theory, 2011

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 value for games with limited communication.

The Shapley value for bicooperative games

2004

and centrA: RESUMEN El objetivo de este trabajo es analizar un concepto de solución que asigna a cada juego bicooperativo un único vector. En el contexto de los juegos bicooperativos introducidos por Bilbao (2000), definimos una solución denominada valor de Shapley porque este valor puede interpretarse de una ma nera semejante al clásico valor de Shapley para juegos cooperativos. El resultado más importante del trabajo es una caracterización axiomática de este valor.

The Shapley value for capacities and games on set systems

2006

We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.

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

Mathematical Social Sciences, 2008

The main focus of this paper is on the restricted Shapley value for multi-choice games introduced by Derks and Peters . 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.

Algorithms for computing the Shapley value of cooperative games on lattices

Discrete Applied Mathematics

We study algorithms to compute the Shapley value for a cooperative game on a lattice L Σ = (F Σ , ⊆) where F Σ is the family of closed sets given by an implicational system Σ on a set N of players. The first algorithm is based on the generation of the maximal chains of the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|Ch|) time complexity using polynomial space, where Ch is the set of maximal chains of L Σ. The second algorithm proceeds by building the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|F Σ |) time and space complexity. Our main contribution is to show that the Shapley value of weighted graph games on a product of chains with the same fixed length is computable in polynomial time. We do this by partitioning the set of feasible coalitions relevant to the computation of the Shapley value into equivalence classes in such a way that we need to consider only one element of each class in the computation.