We propose a novel definition of Shapley values with uncertain value functions based on first principles using probability theory. Such can arise in the context explainable machine learning as result non-deterministic algorithms. show that random effects fact be absorbed into noiseless but shifted function. Hence, used analogy to regular values. However, their reliable evaluation typically requ...

The core and the Shapley value are important solution concepts in cooperative game theory. While the core is designed for the stability of the game, the Shapley value aims for fairness among the players. However, the Shapley value might not lie within the core and a core solution might not be ‘fair’. We introduce a new solution concept called the ‘fairest core’, one that aims for both stability...

This paper makes the case for using Shapley value to quantify the importance of random input variables to a function. Alternatives based on the ANOVA decomposition can run into conceptual and computational problems when the input variables are dependent. Our main goal here is to show that Shapley value removes the conceptual problems. We do this with some simple examples where Shapley value lea...

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some player’s marginal contribution to all coalitions increase or stay the same, then this player’s payoff should not decrease. We investigate the class of values that satisfy effi ciency, symmetry, and weak monotonicity. It turns out that this class coincides with...

We show that the Owen value for TU games with a cooperation structure extends the Shapley value in a consistent way. In particular, the Shapley value is the expected Owen value for all symmetric distributions on the partitions of the player set. Similar extensions of the Banzhaf value do not show this property. Journal of Economic Literature Classification Number: C71.

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...

A well known and simple game to model markets is the glove game where worth is produced by building matching pairs. For glove games, diff erent concepts, like the Shapley value, the restricted Shapley value or the Owen value, yield diff erent distributions of worth. Moreover, computational eff ort of these values is in general very high. This paper provides effi cient allocation formulas of the...

The Shapley Value is arguably the most important normative solution concept in coalitional games. One of its applications is in the domain of networks, where the Shapley Value is used to measure the relative importance of individual nodes. This measure, which is called node centrality, is of paramount significance in many real-world application domains including social and organisational networ...