Cooperative games on antimatroids

Abstract

The aim of this paper is to introduce cooperative games with a feasible coalition system which is called antimatroid. These combinatorial structures generalize the permission structures, which have nice economical applications. With this goal, we …rst characterize the approaches from a permission structure with special classes of antimatroids. Next, we use the concept of interior operator in an antimatroid and we de…ne the restricted game taking into account the limited possibilities of cooperation determined by the antimatroid. These games extend the restricted games obtained by permission structures. Finally, we provide a computational method to obtain the Shapley and Banzhaf values of the players in the restricted game, by using the worths of the original game.

On hierarchies and communication

“…The conjunctive feasible set of any acyclic permission structure is a normal antimatroid such that every player i ∈ N has a unique i-path in F. 4 Clearly, this property is not satisfied by all disjunctive feasible sets as can be seen from Example 2.1 where {1, 2, 4} and {1, 3, 4} are both 4-paths in Φ d D . Further Algaba, Bilbao, van den Brink and Jiménez Losada (2004) show that the disjunctive feasible set of any acyclic permission structure is a normal antimatroid such that deleting the unique endpoint of any path leaves behind a feasible coalition that is again a path. This property is not satisfied by all conjunctive feasible sets as can be seen from Example 2.1 where {1, 2, 3, 4} is the unique 4-path in Φ c D , but {1, 2, 3} is not a path.…”

“…Besides showing that for every permission structure (N, D) it holds that Φ c D and Φ d D are normal antimatroids, Algaba, Bilbao, van den Brink and Jiménez Losada (2004) also characterize those antimatroids that can be the conjunctive or disjunctive feasible set of some acyclic permission structure. They use the following notions for a feasible set F ⊆ 2 N which generalize the concept of a directed path in a permission structure.…”

“…Since we are in the field of restricted cooperation, the answer must be found in the properties of the feasible set. Algaba, Bilbao, van den Brink and Jiménez-Losada (2004) show that the conjunctive and disjunctive feasible sets are antimatroids being sets of feasible coalitions satisfying the following properties (see, Dilworth (1940) and Edelman and Jamison (1985)). A set of feasible coalitions F ⊆ 2 N satisfies accessibility if every nonempty feasible coalition has at least one player that can leave the coalition leaving behind a feasible subcoalition, i.e.…”

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

“…Also notice that every antimatroid is a union closed system by definition. Also the collection of conjunctive feasible coalitions of a permission structure is union closed (see Gilles et al 1992) and this collection is an antimatroid when the permission structure is acyclic (see Algaba et al 2004). …”

“…2 An example of an antimatroid is an acyclic permission structure where players need permission from (some of) their superiors in a hierarchical structure when they want cooperate with others. Since the concept of union closed system is more general than the notion of antimatroid, games on union closed systems are more general than the games on antimatroids as considered in Algaba et al (2003Algaba et al ( , 2004, and, therefore, also more general than the games with acyclic permission structure, considered in Gilles et al (1992), van den Brink and Gilles (1996), Gilles and Owen (1994) and van den Brink (1997).…”