QuadraSHAP: Stable and Scalable Shapley Values for Product Games via Gauss-Legendre Quadrature

• URL: http://arxiv.org/abs/2605.05870v1
• Date: Thu, 07 May 2026 08:36:50 GMT
• Title: QuadraSHAP: Stable and Scalable Shapley Values for Product Games via Gauss-Legendre Quadrature
• Abstract summary: We study the efficient computation of Shapley values for emphproduct games -- cooperative games in which the coalition value factorizes as a product of per-player terms.<n>Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional representation.
• Score: 10.640152986359668
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

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