  Shapley values, which were originally designed to assign attributions to
individual players in coalition games, have become a commonly used approach in
explainable machine learning to provide attributions to input features for
black-box machine learning models. A key attraction of Shapley values is that
they uniquely satisfy a very natural set of axiomatic properties. However,
extending the Shapley value to assigning attributions to interactions rather
than individual players, an interaction index, is non-trivial: as the natural
set of axioms for the original Shapley values, extended to the context of
interactions, no longer specify a unique interaction index. Many proposals thus
introduce additional less ''natural'' axioms, while sacrificing the key axiom
of efficiency, in order to obtain unique interaction indices. In this work,
rather than introduce additional conflicting axioms, we adopt the viewpoint of
Shapley values as coefficients of the most faithful linear approximation to the
pseudo-Boolean coalition game value function. By extending linear to
$\ell$-order polynomial approximations, we can then define the general family
of faithful interaction indices. We show that by additionally requiring the
faithful interaction indices to satisfy interaction-extensions of the standard
individual Shapley axioms (dummy, symmetry, linearity, and efficiency), we
obtain a unique Faithful Shapley Interaction index, which we denote Faith-Shap,
as a natural generalization of the Shapley value to interactions. We then
provide some illustrative contrasts of Faith-Shap with previously proposed
interaction indices, and further investigate some of its interesting algebraic
properties. We further show the computational efficiency of computing
Faith-Shap, together with some additional qualitative insights, via some
illustrative experiments.
