  Complex black-box machine learning models are regularly used in critical
decision-making domains. This has given rise to several calls for algorithmic
explainability. Many explanation algorithms proposed in literature assign
importance to each feature individually. However, such explanations fail to
capture the joint effects of sets of features. Indeed, few works so far
formally analyze high-dimensional model explanations. In this paper, we propose
a novel high dimension model explanation method that captures the joint effect
of feature subsets.
  We propose a new axiomatization for a generalization of the Banzhaf index;
our method can also be thought of as an approximation of a black-box model by a
higher-order polynomial. In other words, this work justifies the use of the
generalized Banzhaf index as a model explanation by showing that it uniquely
satisfies a set of natural desiderata and that it is the optimal local
approximation of a black-box model.
  Our empirical evaluation of our measure highlights how it manages to capture
desirable behavior, whereas other measures that do not satisfy our axioms
behave in an unpredictable manner.
