The Shapley value (Shapley (1953)) is one of the most prominent one-point solution concepts in cooperative game theory that divides revenues (or cost, power) that can be obtained by cooperation of players in the game. The Shapley value is mathematically characterized by properties that have appealing real-world interpretations and hence its use in practical settings is easily justified. The dow...

The Shapley value assigns, to each game that is adequately represented by its characteristic function, an outcome for each player. An elaboration on the Shapley value that assigns, to characteristic function games, a “partition function”outcome is broadly established and accepted, but elaborations to encompass games with externalities (represented by partition functions) are not. Here, I show t...

The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (J Math Biol 67:1163-1170, 2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley valu...

Feature selection is a popular topic. The main approaches to deal with it fall into the three categories of filters, wrappers and embedded methods. Advancement in algorithms, though proving fruitful, may be not enough. We propose integrate an explainable AI approach, based on Shapley values, provide more accurate information for feature selection. test our proposal real setting, which concerns ...

The purpose of this paper is to extend the following result by Monderer et al. (1992): the set of weighted Shapley values includes the Core in TU games. We extend the result to the class of uniformly positively smooth NTU games. We focus on two solution concepts which extend the weighted Shapley value. First, we focus on the weighted Egalitarian solution introduced by Kalai and Samet (1985). We...

We investigate the distribution of the well-studied Shapley–Shubik values in weighted voting games where the agents are stochastically determined. The Shapley-Shubik value measures the voting power of an agent, in typical collective decision making systems. While easy to estimate empirically given the parameters of a weighted voting game, the Shapley values are notoriously hard to reason about ...

Interval Shapley value (ISV) was initially considered for static one-shot games in [1, 2]. In [3, 4] it was shown that the Shapley value in dynamic games is timeinconsistent. To satisfy time-consistency condition for the Shapley value it is necessary to introduce a new control variable, so-called imputation distribution procedure (IDP). The case with interval Shapley value for dynamic games was...

In this paper, we first introduce the model of games on augmenting systems with a coalition structure, which can be seen as an extension of games on augmenting systems. The core of games on augmenting systems with a coalition structure is defined, and an equivalent form is discussed. Meantime, the Shapley function for this type of games is given, and two axiomatic systems of the given Shapley f...

According to the work of Faigle [3] a static Shapley value for games on matroids has been introduced in Bilbao, Driessen, JiménezLosada and Lebrón [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 ...