Additive Self-Attribution (ASA) Overview

• ASA is a framework that decomposes changes in multivariate outputs into additive contributions based on cooperative game theory principles.
• The method rigorously adheres to axioms—Dummy, Additivity, Conditional Nonnegativity, Affine Scale Invariance, and Anonymity—to ensure fairness and consistency.
• Its efficient computation via the Aumann–Shapley–Shubik method enables practical applications in economics, finance, and marketing by clearly attributing influences.

Additive Self-Attribution (ASA) formalizes the decomposition of the output of a multivariate function or model into additive contributions from each input, such that the total change or prediction is entirely explained by the sum of these contributions. The ASA paradigm is rigorously grounded in the principles of cost-sharing from cooperative game theory and is characterized by strict adherence to axioms that ensure fairness, interpretability, and efficiency in attribution. The theoretical foundation, efficient computation for specific function classes, practical applications, and comparison to alternative attribution methods together define the landscape of ASA approaches.

1. Theoretical Foundations and Formalization

Additive Self-Attribution addresses the attribution problem: given a function $f$ defined over $n$ variables and two instances $r$ (reference) and $s$ (target), how can the overall change $f(s) - f(r)$ be decomposed into contributions from each variable? Formally, ASA seeks a vector $(z_1(r, s, f), \dots, z_n(r, s, f))$ such that:

$\sum_{i=1}^n z_i(r, s, f) = f(s) - f(r)$

This framing draws directly on cost-sharing models from cooperative game theory (notably the analysis of characteristic functions), ensuring that the solution possesses desirable properties such as fairness and consistency.

Central to ASA is its axiom-driven construction:

• Dummy: If $f$ is independent of $i$, $z_i(r, s, f) = 0$.
• Additivity: For $f = f_1 + f_2$, $z_i(r, s, f) = z_i(r, s, f_1) + z_i(r, s, f_2)$.
• Conditional Nonnegativity: For $f$ non-decreasing in $x_i$ over $[r_i, s_i]$, a variable's increasing (decreasing) value yields a non-negative (non-positive) attribution.
• Affine Scale Invariance: Attributions are invariant under affine transformations of the variables.
• Anonymity: Attributions are symmetric under permutations of variable orderings.

The “self-attribution” in ASA refers to each variable receiving a share of the output or change that is consistent with these axioms and hence can be linearly summed to recover the original outcome.

2. The Aumann–Shapley–Shubik Attribution Method

For functions comprising sums of multilinear and additive components, there exists a unique attribution method—termed the Aumann–Shapley–Shubik method—that satisfies all above axioms (Sun et al., 2011). The quantitative assignment for variable $i$ can be computed either as:

• Aumann–Shapley (Path Integration) Form:

$$z_i(r, s, f) = \int_0^1 \frac{\partial f}{\partial x_i}(r + t(s - r)) \cdot (s_i - r_i) \, dt$$

integrating the marginal effect along the straight line from $r$ to $s$.

• Shapley–Shubik (Combinatorial) Form:

$$z_i(r, s, f) = (s_i - r_i) \cdot \frac{1}{n!} \sum_{J \subseteq [n] \setminus \{i\}} |J|! (n-1-|J|)! \left( \prod_{j \in J} s_j \right) \left( \prod_{j \in [n] \setminus (\{i\} \cup J)} r_j \right)$$

For multilinear $f$, these two forms coincide, providing both a continuous and discrete route to compute theoretically consistent attributions.

For the bilinear case $f(r_1, r_2) = r_1 r_2$:

$$z_1(r, s, f) = (s_1 - r_1) \frac{r_2 + s_2}{2}\quad,\quad z_2(r, s, f) = (s_2 - r_2) \frac{r_1 + s_1}{2}$$

3. Multilinear Characteristic Functions and Computational Efficiency

A function $f$ is multilinear if it is a sum of monomials, each of at most first degree in each variable:

$$f(r_1, ..., r_n) = \sum_{I \subseteq [n]} c_I \prod_{i \in I} r_i$$

ASA is theoretically tractable for such functions, as the interplay between multilinearity and the attribution axioms ensures the path integral reduces to the combinatorial sum, allowing for efficient computation:

• For $f(r) = r_1 \dots r_n$, Theorem 4.13 demonstrates that $z_i(r, s, f)$ is computable in $O(n^2)$ time and $O(n)$ memory (Sun et al., 2011).

Multilinear or multilinear-plus-additive structures are common in economic, financial, and production models, where interpretability and computational feasibility are critical.

4. Practical Applications

ASA-driven attributions are directly applicable to a range of high-dimensional analysis settings:

Domain	Function Form	ASA Role
Pay-per-click advertising	$s = c \cdot p$	Decomposes spend into click/cost changes
Portfolio analysis	$\sum_i r_i w_i$	Separates allocation from selection effect

In these cases, ASA (via the Aumann–Shapley–Shubik method) ensures that decomposition of changes in aggregate metrics reflects the true influence of component variables or decisions. The attribution thus informs resource allocation, campaign optimization, and risk management by assigning responsibility for outcomes to underlying factors.

5. Comparison with Alternative Attribution Schemes

Many alternative attribution approaches exist, such as Taylor expansions at a point or “path” integrations along arbitrary trajectories. However, only the Aumann–Shapley–Shubik method uniquely satisfies the combined axioms for the specified function class (Sun et al., 2011).

Key distinctions include:

• Uniqueness: Agreement between path-based (Aumann–Shapley) and permutation-averaged (Shapley–Shubik) attributions holds only for multilinear-plus-additive functions. Outside this class, these methods disagree (see Theorem 4.16).
• Axiomatic Guarantees: Most other methods violate at least one critical axiom (such as Affine Scale Invariance or Anonymity) for generic function classes.
• Computational Practicality: ASA is both efficient and exact for multilinear functions, whereas alternative approaches may be computationally demanding or yield less interpretable decompositions.

ASA further ensures completeness and Conditional Nonnegativity (a variable that increases monotonically receives a non-negative attribution), properties not automatically enforced in alternative schemes.

6. Broader Significance and Theoretical Implications

The ASA framework, exemplified by the Aumann–Shapley–Shubik method, provides a structured, axiomatic approach to attribution that bridges cooperative game theory and applied functional analysis. By grounding attribution in fairness, additivity, and invariance, ASA offers a rigorous blueprint for decomposing changes in complex multivariate functions. The matching of continuous path and discrete order-based attributions is both of intrinsic mathematical interest and practical utility.

This framework is extensible to cost-sharing, resource allocation, multichannel attribution in marketing, and risk analysis in finance. Where multilinear or sum-of-multilinear-plus-additive models are appropriate, ASA guarantees both interpretability and computational tractability, distinguishing it sharply from more ad hoc or computationally intensive attributions.

7. Summary

Additive Self-Attribution defines a unique and principled solution to the attribution problem for multilinear and related function classes, unifying the continuous and combinatorial perspectives of cost-sharing. The resulting attributions are fair, interpretable, and efficiently computable, with widespread applicability in economics, finance, and data-driven disciplines. The theoretical foundations and practical relevance of ASA position it as a central tool for both attribution analysis and the design of fair, transparent explanatory models (Sun et al., 2011).