Nonlinear Attribution Improves Player Ranking in Cooperative Games.

Weida Li, Zhuanghua Liu, Yaoliang Yu, Bryan Kian Hsiang Low· July 14, 2026 View original

Key takeaways

  • The linear Shapley value can be unreliable for accurately ranking player contributions due to its large null space.
  • Nonlinear axiomatic attribution methods offer a more effective alternative for identifying positive contributions.
  • These new methods retain necessary axioms while improving performance on metrics like inclusion AUC.
  • They provide a more faithful approximation of utility functions in cooperative games.

Who benefits

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Summary

This paper introduces a class of nonlinear axiomatic attribution methods for cooperative games, addressing the limitations of the linear Shapley value in accurately ranking player contributions. These new methods offer improved performance in identifying positively participating players, especially when evaluated by the inclusion AUC metric.

The Shapley value is a cornerstone in attribution problems, known for satisfying key axioms like linearity and efficiency. However, this research highlights a critical flaw: its linearity can make it unreliable for ranking player contributions, particularly when evaluated using the inclusion AUC metric. The issue stems from the Shapley value acting as a linear operator with a large null space, meaning it can't distinguish between certain non-negligible perturbations in player contributions. To overcome this, the paper proposes a new class of nonlinear axiomatic attribution methods. Drawing inspiration from the least core, a known nonlinear alternative to the Shapley value, these methods are designed to retain essential axioms while introducing nonlinearity. Each method yields a unique contribution vector by solving a minimization problem that aims to approximate utility functions more faithfully. Experimental results demonstrate that these nonlinear methods can be more effective than Shapley value variants (even those relaxing the efficiency axiom) in terms of the inclusion AUC metric. This suggests they provide a more accurate and reliable way to identify and rank the positive contributions of players in cooperative game theory settings, which has implications for various real-world attribution scenarios.

Why it matters

Professionals needing to fairly attribute credit or responsibility in complex systems, such as feature importance in machine learning, team contributions, or resource allocation, can benefit from more accurate nonlinear attribution methods. This can lead to better decision-making and resource optimization.

How to implement this in your domain

  1. 1Evaluate the limitations of the Shapley value in your current attribution problems, especially if player ranking is critical.
  2. 2Explore the proposed nonlinear attribution methods as alternatives for more accurate contribution assessment.
  3. 3Implement and test these nonlinear methods using the provided code or by developing custom solutions.
  4. 4Compare the performance of nonlinear attribution against traditional Shapley values using metrics like inclusion AUC in your specific domain.

Original post by Weida Li, Zhuanghua Liu, Yaoliang Yu, Bryan Kian Hsiang Low

"arXiv:2607.09869v1 Announce Type: new 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 p…"

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Originally posted by Weida Li, Zhuanghua Liu, Yaoliang Yu, Bryan Kian Hsiang Low on X · view source

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