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...

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 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.