Shapley-Taylor Index: Interaction Attribution

• Shapley-Taylor Index is an interaction-attribution method that extends the Shapley value to capture joint effects of feature subsets up to a chosen order k.
• It employs a truncated Taylor series on a set function's multilinear extension, distinguishing main effects from pairwise and higher-order interactions.
• The method ensures the total prediction difference is conserved while unresolved higher-order effects are absorbed into the top-order interaction bucket.

Searching arXiv for recent and foundational papers on the Shapley-Taylor Index and closely related interaction-attribution work. The Shapley-Taylor Index is an interaction-attribution method that extends the Shapley value from singleton features to subsets of features up to a chosen order $k$. For a set function $F:2^N\to\mathbb{R}$, it distributes the prediction difference $F(N)-F(\emptyset)$ across all subsets $S\subseteq N$ with $|S|\le k$, treating singleton terms as main effects, subsets of size $2,\dots,k-1$ as lower-order interactions, and subsets of size $k$ as highest-order terms that also absorb interactions of order $>k$. The method is named for its combination of Shapley-style random-order averaging with a truncated Taylor expansion of the multilinear extension of the set function (Dhamdhere et al., 2019). Later work places it within a broader family of Shapley-based interaction explanations and interprets it as an order-controlled truncation of an underlying generalized additive decomposition induced by the choice of value function (Bordt et al., 2022).

1. Problem setting and conceptual scope

The index was introduced to address a limitation of ordinary Shapley values: standard feature attribution assigns all explanatory mass to singleton features, even when the prediction depends primarily on joint effects. In models containing terms such as $x_1x_2x_3$, ordinary Shapley values divide the gain among the participating features but do not explicitly represent the interaction as such. The Shapley-Taylor construction instead attributes a prediction to subsets of features up to a specified order $k$, so that explanations can contain main effects, pairwise interactions, triple interactions, and so forth, subject to the truncation order (Dhamdhere et al., 2019).

In the foundational formulation, the model is represented as a set function

$F:2^N\to\mathbb{R}$0

where $F:2^N\to\mathbb{R}$1 is the feature set and $F:2^N\to\mathbb{R}$2 is the model output when exactly the features in $F:2^N\to\mathbb{R}$3 are present and the others are ablated or baselined. The central attribution problem is then to distribute

$F:2^N\to\mathbb{R}$4

over subsets $F:2^N\to\mathbb{R}$5. The Shapley-Taylor index assigns values $F:2^N\to\mathbb{R}$6 to all subsets with $F:2^N\to\mathbb{R}$7. The interpretation is order-sensitive: if $F:2^N\to\mathbb{R}$8, the term is a main effect; if $F:2^N\to\mathbb{R}$9, it is a lower-order interaction; if $F(N)-F(\emptyset)$0, it is the highest represented interaction order and also serves as a remainder-like bucket for effects of order larger than $F(N)-F(\emptyset)$1 (Dhamdhere et al., 2019).

Two conceptual points are central. First, when $F(N)-F(\emptyset)$2, the construction reduces exactly to the ordinary Shapley value. Second, truncation is deliberate rather than incidental: the top-order layer is designed to absorb unresolved higher-order structure instead of forcing all such structure into singleton attributions. This distinction later becomes central in the generalized additive model interpretation, where ordinary Shapley values appear as the $F(N)-F(\emptyset)$3 case of a broader order-truncated family (Bordt et al., 2022).

2. Formal definition

The formalism is built from discrete set-function derivatives. For $F(N)-F(\emptyset)$4,

$F(N)-F(\emptyset)$5

and for $F(N)-F(\emptyset)$6,

$F(N)-F(\emptyset)$7

More generally, for $F(N)-F(\emptyset)$8,

$F(N)-F(\emptyset)$9

which is the general interaction primitive used by the method (Dhamdhere et al., 2019).

Let

$S\subseteq N$0

be a permutation of the features, and let $S\subseteq N$1 denote the set of predecessors of $S\subseteq N$2 in $S\subseteq N$3. For a subset $S\subseteq N$4, define

$S\subseteq N$5

the set of features that appear before every element of $S\subseteq N$6 in the permutation. The permutation-specific Shapley-Taylor contribution is

$S\subseteq N$7

The Shapley-Taylor index is then the expectation over a uniformly random permutation: $S\subseteq N$8 For $S\subseteq N$9, the quantity is already permutation-independent, so

$|S|\le k$0

For $|S|\le k$1, the paper derives the closed form

$|S|\le k$2

Thus the complete definition is

$|S|\le k$3

This formulation makes explicit that lower-order terms are evaluated at the baseline, whereas top-order terms average discrete interaction increments over contexts of all sizes (Dhamdhere et al., 2019).

3. Taylor-series interpretation and axiomatic characterization

A defining result of the original paper is that the index is exactly the truncated Taylor expansion of the multilinear extension of the set function. The multilinear extension is

$|S|\le k$4

with $|S|\le k$5. Along the diagonal path

$|S|\le k$6

one has $|S|\le k$7 and $|S|\le k$8. Writing $|S|\le k$9 for the mixed partial derivative indexed by $2,\dots,k-1$0, the truncated Taylor expansion with Lagrange remainder yields

$2,\dots,k-1$1

The exact correspondence theorem states that for $2,\dots,k-1$2,

$2,\dots,k-1$3

and for $2,\dots,k-1$4,

$2,\dots,k-1$5

Lower-order Shapley-Taylor terms are therefore literal Taylor coefficients at the baseline, while the order-$2,\dots,k-1$6 terms come from the remainder term (Dhamdhere et al., 2019).

The method is also given an axiomatic foundation. It is characterized as the unique interaction index satisfying linearity, dummy, symmetry, efficiency, and an additional interaction distribution axiom. Efficiency takes the form

$2,\dots,k-1$7

The interaction distribution axiom governs how pure higher-order interactions are allocated under truncation. For an interaction function $2,\dots,k-1$8, it requires that for all $2,\dots,k-1$9 with $k$0,

$k$1

This prevents lower-order terms from absorbing interaction mass that belongs to larger coalitions, and forces unresolved higher-order effects into the highest represented order $k$2 (Dhamdhere et al., 2019).

On unanimity functions $k$3 iff $k$4, the characterization becomes especially transparent. If $k$5, the interaction is represented exactly at its own order. If $k$6, the mass is evenly distributed among the $k$7-subsets of $k$8, each receiving

$k$9

This basis-level behavior is the algebraic core of the truncation rule (Dhamdhere et al., 2019).

4. Relation to other interaction indices and to functional decomposition

The principal contrast in the original paper is with the older Shapley Interaction index from cooperative game theory,

$>k$0

The key difference is not the use of discrete derivatives, which both methods share, but the allocation of higher-order interaction mass across orders. Shapley-Taylor satisfies efficiency for explanations up to order $>k$1, whereas the Shapley Interaction index does not satisfy that efficiency property in the same truncated-explanation sense. The original paper argues that this can cause the Shapley Interaction index to overstate interactions when used for interpretability (Dhamdhere et al., 2019).

A canonical example is

$>k$2

For pairwise explanation ($>k$3), Shapley-Taylor gives main effects $>k$4 and pairwise interactions $>k$5 for each pair, so the total interaction effect is $>k$6. By contrast, the Shapley Interaction index gives pairwise interactions $>k$7 each, for a total interaction effect $>k$8 (Dhamdhere et al., 2019). The majority-function example in the same paper makes the same point differently: pairwise Shapley-Taylor singleton terms are $>k$9 for $x_1x_2x_3$0, while pairwise interaction terms are equal and sum to $x_1x_2x_3$1, so the function is represented as entirely interaction-driven (Dhamdhere et al., 2019).

Later work recasts the Shapley-Taylor index within a broader local-explanation framework built from a value function $x_1x_2x_3$2 for a prediction function $x_1x_2x_3$3. Under subset compliance, any such value function induces a unique functional decomposition

$x_1x_2x_3$4

with

$x_1x_2x_3$5

where

$x_1x_2x_3$6

This is the full Möbius or Harsanyi decomposition, called the Shapley-GAM because it links Shapley-style value functions to generalized additive model representations (Bordt et al., 2022).

Within this framework, the order-$x_1x_2x_3$7 Shapley-Taylor index satisfies

$x_1x_2x_3$8

Hence the order-$x_1x_2x_3$9 index recovers the true component $k$0 exactly for all lower-order subsets $k$1, while the top-order layer $k$2 absorbs higher-order terms with weights $k$3. This is the functional-decomposition interpretation of truncation: interactions through order $k$4 are represented explicitly, and all remaining structure is compressed into the highest admissible order rather than redistributed throughout all lower orders (Bordt et al., 2022).

A central recovery theorem states that if $k$5 is a generalized additive model of order $k$6, then under interventional SHAP, or under observational SHAP with independent features, $k$7-Shapley values, the Shapley-Taylor index of order $k$8, and the Faith-Shap interaction index of order $k$9 all recover the same GAM representation: $F:2^N\to\mathbb{R}$00 In that regime, Shapley-Taylor does not merely provide a local attribution approximation; it exactly identifies the additive and interaction components of the model (Bordt et al., 2022).

5. Computation, approximation, and estimation issues

Exact computation is exponential in the number of features because the index is defined by subset summation or, equivalently, by averaging over permutations while evaluating discrete derivatives. The original paper states worst-case complexity as

$F:2^N\to\mathbb{R}$01

and describes two practical approximations: permutation sampling, analogous to sampling-based Shapley estimation, and feature preselection, in which a smaller set of important features is identified first and interaction analysis is then restricted to that subset (Dhamdhere et al., 2019).

The generalized-additive interpretation paper makes the same computational point from the value-function side. There, observational and interventional value functions involve expectations, and the authors state that they “simply evaluate the value function for all possible subsets $F:2^N\to\mathbb{R}$02, then combine the respective terms according to Definition \ref{def:n_shapley_values}.” They provide the Python package nshap for computing $F:2^N\to\mathbb{R}$03-Shapley values as well as Shapley Taylor- and Faith-Shap interaction indices, and note that approximation methods analogous to Kernel SHAP should be possible (Bordt et al., 2022).

A later unification result treats the Shapley-Taylor Interaction Index as a member of the class of cardinal interaction indices (CII), defined generally as

$F:2^N\to\mathbb{R}$04

That paper states that STI satisfies linearity, symmetry, and dummy, and therefore falls under the CII representation theorem. For top-order STI, the appendix gives the explicit weight

$F:2^N\to\mathbb{R}$05

and shows that SHAP-IQ can be instantiated directly for STI by plugging this weight into the general estimator (Fumagalli et al., 2023).

The same work proves generic guarantees for any CII, which therefore apply to STI: unbiasedness,

$F:2^N\to\mathbb{R}$06

consistency,

$F:2^N\to\mathbb{R}$07

and the error bound

$F:2^N\to\mathbb{R}$08

It also proves an STI-specific structural statement: SII and STI are $F:2^N\to\mathbb{R}$09-efficient, and SHAP-IQ estimates maintain efficiency for STI (Fumagalli et al., 2023).

Two estimation caveats recur across the literature. First, all such explanations depend on the chosen value function or game semantics; different definitions of coalition value induce different interaction attributions. Second, precise estimation of high-order interaction terms requires many samples. In the checkerboard experiments in the generalized-additive paper, high-order interactions can in principle be detected, but precise estimation requires many samples for the value function, and insufficient sampling can introduce spurious lower- or intermediate-order effects (Bordt et al., 2022).

6. Extensions, structured variants, and applications

Subsequent work has adapted the Shapley-Taylor idea to settings in which the original unrestricted coalition semantics are too coarse or structurally mismatched. In graph explanation, the Myerson-Taylor interaction index replaces the original set function $F:2^N\to\mathbb{R}$10 by the graph-restricted function

$F:2^N\to\mathbb{R}$11

where $F:2^N\to\mathbb{R}$12 is the set of connected components of the subgraph induced by $F:2^N\to\mathbb{R}$13. The resulting interaction rule is

$F:2^N\to\mathbb{R}$14

This paper states explicitly that

$F:2^N\to\mathbb{R}$15

so Myerson-Taylor is Shapley-Taylor applied to a graph-restricted game. It is characterized as the unique index satisfying Linearity, Restricted Null Player, Coalitional Fairness, Interaction Distribution, and Component Efficiency, and reduces to ordinary Shapley-Taylor when the graph is complete (Bui et al., 2024).

In multimodal retrieval, the Shapley-Taylor idea has also been repurposed as an internal interaction-learning signal rather than a post hoc explainer. The Pyramidal Shapley-Taylor learning framework uses pairwise, order-2 STI between motion tokens and text tokens,

$F:2^N\to\mathbb{R}$16

with $F:2^N\to\mathbb{R}$17 instantiated as a retrieval similarity score. Because exact permutation averaging is too expensive during training, the method uses Monte Carlo STI targets and a learned STI Estimation Head with KL distillation. The paper explicitly sets $F:2^N\to\mathbb{R}$18, so its use of Shapley-Taylor is pairwise and approximate rather than a full exact implementation of the classical index (Chen et al., 29 Jan 2026).

A separate line of work on graph-structured tensor-network surrogates is mathematically adjacent even when it does not explicitly compute Shapley-Taylor itself. That framework learns a multilinear surrogate for the coalition game and then recovers higher-order Shapley-style interaction quantities from mixed partial derivatives along the diagonal of the multilinear extension. Its main explicit interaction formalism is the Shapley interaction index rather than STI, but the paper states that a broad class of indices, including “Shapley--Taylor / Shapley interaction families,” can be written as linear functionals of the same multilinear extension (Heidari et al., 1 Jun 2026).

These developments clarify the present status of the Shapley-Taylor Index. It remains the canonical truncation-based interaction extension of the Shapley value: a method that preserves efficiency, protects lower-order terms from contamination by higher-order interaction mass, and admits both a Taylor-series interpretation and a generalized-additive decomposition view. At the same time, its semantics remain definition-dependent: the index explains a chosen coalition game or value function, and later work repeatedly emphasizes that the choice of value function, masking scheme, or structural restriction is not secondary but constitutive of what the interaction scores mean (Dhamdhere et al., 2019).