  We compare machine learning explainability methods based on the theory of
atomic (Shapley, 1953) and infinitesimal (Aumann and Shapley, 1974) games, in a
theoretical and experimental investigation into how the model and choice of
integration path can influence the resulting feature attributions. To gain
insight into differences in attributions resulting from interventional Shapley
values (Sundararajan and Najmi, 2019; Janzing et al., 2019; Chen et al., 2019)
and Generalized Integrated Gradients (GIG) (Merrill et al., 2019) we note
interventional Shapley is equivalent to a multi-path integration along $n!$
paths where $n$ is the number of model input features. Applying Stoke's theorem
we show that the path symmetry of these two methods results in the same
attributions when the model is composed of a sum of separable functions of
individual features and a sum of two-feature products. We then perform a series
of experiments with varying degrees of data missingness to demonstrate how
interventional Shapley's multi-path approach can yield less consistent
attributions than the single straight-line path of Aumann-Shapley. We argue
this is because the multiple paths employed by interventional Shapley extend
away from the training data manifold and are therefore more likely to pass
through regions where the model has little support. In the absence of a more
meaningful path choice, we therefore advocate the straight-line path since it
will almost always pass closer to the data manifold. Among straight-line path
attribution algorithms, GIG is uniquely robust since it will still yield
Shapley values for atomic games modeled by decision trees.
