The Shapley value for games on matroids: The dynamic model

Abstract

Abstract. 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. In this new model for games on matroids, our main result is that there exists a unique value satisfying analogous axioms to the classical Shapley value. Moreover, we obtain a recursive formula to calculate this dynamic Shapley value. Finally, we prove that its components are probabilistic values. Mathematics Subject Classification 2000: 91A12

The fuzzy core and Shapley function for dynamic fuzzy games on matroids

“…From Theorem 3.4, we obtain Bilbao et al (2002) proposed the Shapley function for dynamic crisp games on matroids, and proved the uniqueness of the given Shapley function by linearity, the substitution for unanimity games, dummy property and dynamic efficiency. In the section, we shall discuss the Shapley function for dynamic fuzzy games on matroids, and give the unique proof of the given Shapley function usingdynamic efficiency, symmetry andadditivity.…”

“…From Lemma 2.1 in Bilbao et al (2002), we get the relationship between the probability distribution of fuzzy games on matriods and the probability distribution of dynamic fuzzy games on matriods as follows: …”

“…We first give the Shapley function, proposed by Bilbao et al (2002), for dynamic crisp games. Let (N , M) be a matroid in N , and ν 0 ∈ G 0 (N ).…”

“…Bilbao et al (2001) introduced matroids to game theory, and discussed the Shapley function for games on matroids, which can be considered as a kind of games with coalition structure. Bilbao et al (2002) further discussed the Shapley function for dynamic games on matroids, and gave the unique proof of the given Shapley function usingdynamic efficiency, symmetry property applied to unanimity games, dummy player property and linearity. Sun and Driessen (2003) studied the marginal solution for games on matroids.…”

“…In this paper, following Bilbao et al (2002), we will discuss the fuzzy core and the Shapley function for dynamic fuzzy games on matroids. In general, the characteristic function for fuzzy games is difficultly given.…”