The Shapley value for games on matroids: The static model

Abstract

Abstract. In the classical model of cooperative games, it is considered that each coalition of players can form and cooperate to obtain its worth. However, we can think that in some situations this assumption is not real, that is, all the coalitions are not feasible. This suggests that it is necessary to rise the whole question of generalizing the concept of cooperative game, and therefore to introduce appropriate solution concepts. We propose a model for games on a matroid, based in several important properties of this combinatorial structure and we introduce the probabilistic Shapley value for games on matroids.

The Shapley value for games on matroids: The dynamic model

“…We recall the concept of influence of a feasible coalition for the static model introduced in Bilbao et al [1]. With respect to a given probability distribution P M on BðMÞ, the static influence of S A M is the sum of the probabilities of basic coalitions containing coalition S, i.e.…”

“…Moreover, we provide a recursive formula to calculate the dynamic Shapley value using a sequence of static Shapley values. We now recall some formulas for the static Shapley value obtained in Bilbao et al [1].…”

“…It can occur that the players would only want to study which basic coalition will be formed and how they will share the profits. This is the static model which we have already studied in Bilbao et al [1].…”

“…In addition, we present the relationship between the dynamic Shapley value and the so-called static Shapley value, which has been introduced in Bilbao et al [1] on the basis of four similar axioms, incorporating a slightly adapted efficiency axiom. The proof of the relationship between the dynamic and static Shapley values (applied to so-called unanimity games) is fully based on the relationship between the dynamic and static influence of any feasible coalition.…”

“…According to the work of Faigle [3] a static Shapley value for games on matroids has been introduced in Bilbao, Driessen, Jiménez-Losada and Lebró n [1]. In this paper we present a dynamic Shapley value by using a dynamic model which is based on a recursive sequence of static models.…”