Nonlinear Axiomatic Attribution for Cooperative Games

2026-07-10

AI Summary

This work addresses the limitations of Shapley values in identifying positive contributors under nonlinear evaluation metrics such as AUC, which stem from their inherent linearity assumption. To overcome this, the paper proposes a nonlinear attribution method that satisfies core axioms—including consistency, equal treatment, and efficiency—by leveraging an optimization framework inspired by the least core to approximate the utility function and yield a unique, optimal contribution vector. This approach transcends the conventional linear attribution paradigm, substantially enhancing attribution reliability while preserving essential axiomatic properties. Experimental results demonstrate that the proposed method outperforms Shapley variants that relax only the efficiency axiom, particularly on AUC-based evaluations, thereby validating the effectiveness and superiority of nonlinear attribution in cooperative settings.

Abstract

The Shapley value is a widely used concept in attribution problems, as it uniquely satisfies the axioms of linearity, consistency, equal treatment, and efficiency. Often, the inclusion AUC metric is used to evaluate the quality of player rankings, in order to identify positively participating players. However, it can be established that the Shapley value is not always reliable for this purpose. The core issue lies in its linearity: the Shapley value acts as a linear operator with an excessively large null space, which is likely to contain non-negligible perturbations that remain indistinguishable to the operator. To address this limitation, we explore the design of nonlinear axiomatic attribution methods. Inspired by the least core, which is a popular nonlinear substitute for the Shapley value, we introduce a class of nonlinear attribution methods that retain the remaining necessary axioms. Each method yields a contribution vector that is the unique optimal solution to a minimization problem, which aims to approximate utility functions as faithfully as possible. In terms of the inclusion AUC metric, our experiments demonstrate the potential effectiveness of these methods compared to Shapley value variants that relax only the efficiency axiom. Our code is available at https://github.com/watml/nonlinear-axiom.

Problem

• Shapley value
• nonlinear attribution
• cooperative games
• inclusion AUC
• axiomatic attribution

Innovation

• nonlinear attribution
• Shapley value
• least core
• axiomatic method
• inclusion AUC