Aumann-SHAP: Bridging Shapley Interaction Attribution and Integrated Gradients

Feature attribution can describe a baseline-to-target change through endpoint coalitions or through a continuous path. For interacting features, these views need not divide responsibility in the same way, as endpoint Shapley splits a Harsanyi interaction pot symmetrically, whereas path attribution can reveal asymmetric roles. We introduce Aumann-SHAP, a discrete-to-continuous framework that preserves each endpoint interaction pot while refining how it is allocated. For every interaction set, we isolate a pure interaction surface on the local hypercube, discretize feature changes into micro-moves, and apply Shapley attribution to the induced micro-player game. At resolution $m=1$, Aumann-SHAP recovers equal-split Shapley exactly; under smooth refinement, it converges to the Aumann-Shapley / straight-line Integrated Gradients allocation of the isolated interaction surface. An exact grid-state formulation requires $\mathcal{O}(k(m+1{)}^k)$ operations for a $k$-way interaction. Experiments show that transition geometry can change signs, rankings, and early edit priorities, while often leaving endpoint priorities unchanged. Thus endpoint interaction pots quantify how much interaction is present; Aumann-SHAP additionally captures how responsibility develops between the endpoints.