Shapley Value Distillation

• The paper introduces an algorithm that reconstructs a voting scheme or model to match targeted Shapley value profiles with provable error bounds.
• It employs Fourier analysis and boosting techniques along with anti‐concentration methods to estimate degree-1 correlations from a specialized non-uniform distribution.
• Applications span social choice, cooperative game theory, and explainable AI, providing scalable, approximate solutions to inverse attribution problems.

Shapley value distillation refers to the process of synthesizing, approximating, or reconstructing a function, decision process, or model—in particular, a voting scheme or machine learning model—from a specified vector of Shapley values or related attribution indices, such that the resulting system reflects these desired feature or player attributions. This concept arises most fundamentally in the context of the “inverse Shapley problem” for weighted voting schemes, but the core methods and structural insights have far broader algorithmic and theoretical significance within cooperative game theory, social choice, machine learning, and explainable AI.

1. Formulation of the Inverse Shapley Value Problem

The classical Shapley value, introduced by Shapley and Shubik, quantifies the average marginal contribution (or “power”/“influence”) of each player in a cooperative game, such as a voting scheme. Given a Boolean function $f:\{-1,1\}^n \to \{-1,1\}$ representing a weighted voting scheme, the Shapley value for the $i$th voter is historically derived via combinatorial analysis of coalition orderings.

The inverse Shapley value problem is formulated as follows: Given a target vector $(a_1,a_2,\dots,a_n)$ of Shapley–Shubik indices, construct an explicit weighted voting scheme (equivalently, a monotone linear threshold function f) so that the realized Shapley vector of $f$ approximates $(a_1,\ldots,a_n)$ up to $\epsilon$ in Euclidean distance. This requires a method to “distill” a model or function from its prescribed (local or global) attribution pattern.

Key analytic expressions from (De et al., 2012) foundationally reframe the Shapley value computation in terms of degree-1 expectation under a specially chosen symmetric but non-uniform distribution $\mu$ on $\{-1,1\}^n$. For any Boolean function $f$,

$$\tilde{f}(i)= \frac{f(\mathbf{1}) - f(\mathbf{-1})}{n} + \frac{\Lambda(n)}{2}\left(f^*(i) - \frac{1}{n}\sum_{j=1}^{n} f^*(j)\right),$$

where $f^*(i) = \mathbb{E}_{x \sim \mu}[f(x)x_i]$ and $$\Lambda(n)=\sum_{k=1}^{n-1}\left(\frac{1}{k}+\frac{1}{n-k}\right)$$.

2. Algorithmic Techniques: Reconstruction via Fourier Analysis and Boosting

The algorithm described in (De et al., 2012) proceeds through the following steps for Shapley value distillation:

• Fourier-Analytic Rewriting: The algorithm analyzes the relation between the target Shapley values and the degree-1 Fourier coefficients of the unknown Boolean function under the $\mu$ distribution. Specifically, for each coordinate:

$$\hat{f}(i) = \frac{2\beta}{\Lambda(n)}\left(\tilde{f}(i)-\frac{f(\mathbf{1})-f(\mathbf{-1})}{n}\right)+\frac{1}{n}\sum_{j=1}^{n}\hat{f}(j),$$

where $\beta$ is determined by orthonormality under $\mu$.

• Anti-Concentration Bound: For $f(x)=\mathrm{sign}(w\cdot x-\theta)$ that is monotone and “$\eta$-reasonable” (the threshold $\theta$ is not excessively large in magnitude), $w\cdot x-\theta$ does not concentrate too narrowly under $\mu$. This property is crucial for the stability and learnability of the procedure.
• Boosting-Style Estimation of Correlations: Using a “Boosting–TTV” procedure (Takimoto–Tulsiani–Vempala), the algorithm constructs a bounded function $h$ (LBF: linear bounded function) whose degree-1 correlations with respect to $\mu$ closely match the target values. $h(x)$ has the form:

$$h(x) = P_1\left(\frac{\xi}{2}\sum_{\ell\in\mathcal{L}} w_\ell \ell(x)\right)$$

with $P_1$ the “clipping” to $[-1,1]$.

• Conversion to Linear Threshold Function: After ensuring $\ell_1$ closeness in the Fourier domain and low total error in the $h$–target Shapley vector, $h$ is rounded to a LTF (by taking $\mathrm{sign}(h(x))$) to yield the final distillation.

There are two algorithmic variants:

• Arbitrary Weights Case: Running time poly$(n,2^{\mathrm{poly}(1/\epsilon)})$, Shapley distance $\le \epsilon$.
• Bounded Weights Case: If integer weights up to $W$ exist for some LTF approximately matching the target indices, the algorithm runs in poly$(n,W)$ time and achieves Shapley error $n^{-1/8}$.

3. Structural and Performance Guarantees

The correctness of this distillation method is established by linking the Fourier closeness of $h$ and $f$ to the $\ell_2$ (Euclidean) distance between their Shapley vectors. If a monotone, $\eta$-reasonable LTF exists with Shapley vector $\epsilon$-close to the target, the algorithm guarantees outputting a LTF $h$ such that

$$d_{S}(f,h)=\sqrt{\sum_{i=1}^n (\tilde{f}(i) - \tilde{h}(i))^2} \le \epsilon.$$

In the bounded weights setting (integer weights at most $W$), the guarantee holds for all error thresholds $\epsilon \gg n^{-c}$ with running time polynomial in $n$ and $W$.

This establishes the first existence of a polynomial-time algorithm (in $n$ and $W$) for Shapley value distillation in the inverse setting, with provable approximation bounds as a function of $\epsilon$ (or $n^{-1/8}$).

4. Applications Across Social Choice, Game Theory, and Machine Learning

Distilling a voting scheme or model to match prescribed Shapley values has substantive applications:

• Social Choice and Cooperative Game Theory: Enables the explicit design of weighted voting games or allocation mechanisms in which each player's power (as measured by the Shapley–Shubik index) is close to a specified vector. This is fundamental for fair voting system design, cost allocation, and institutional analysis.
• Machine Learning and Explainable AI: In feature attribution settings, typically the “forward” problem is solved—computing Shapley values for an existing model to explain predictions. Shapley value distillation is the “inverse”: constructing a model whose local/global Shapley attribution profile matches a target, either for benchmarking, driving fairness interventions, auditing, or even for reverse-engineering model structures from observed attributions.
• Foundations for Algorithmic Game Theory & High-Dimensional Learning: The interplay of boosting, Fourier methods, and anti-concentration underlies several algorithmic domains beyond voting, including learning thresholds, inverse problems for Boolean functions, and theoretical guarantees for attributions.

5. Contrasts to Prior Approaches and Broader Methodological Impact

Before (De et al., 2012), methods for the inverse Shapley value problem were restricted to brute-force enumeration, exponential search, or relied on heuristics without performance guarantees. The combination of a tailored non-product symmetric distribution $\mu$, analytic structural reformulation, and boosting-based estimation is novel in the Shapley context. Compared to “Chow parameter” inversion (for Banzhaf indices), the Shapley setting is more subtle due to the role of permutations and the specific non-uniform averaging; the present framework generalizes and strengthens these ideas.

Algorithmic use of boosting methods allows sidestepping the need for exhaustive enumeration, paving the way for scalable distillation and even the possibility of iterative or statistical estimation when the exact target is not available.

6. Assumptions, Limitations, and Future Research

• Assumption of “Reasonableness”: The solution is guaranteed when a $\eta$-reasonable LTF exists—a function not too close to constant. The boundary cases (nearly degenerate thresholds) are not covered and remain open.
• EPTAS, Not FPTAS: The dependence on $1/\epsilon$ in the exponent for the arbitrary weights case is exponential; improving this to a fully polynomial time approximation scheme is a concrete open direction.
• Exact vs Approximate Inversion: Approximate distillation is tractable with guarantees; the exact inverse Shapley problem (finding a voting scheme matching a target vector exactly) is likely #P-hard.

Additional unresolved challenges include extending the approach when the weights are required to be strictly integer, relaxing monotonicity or “reasonableness,” and integrating the approach with data-driven or statistical estimation scenarios where only noisy empirical Shapley profiles are available.

7. Broader Implications and Extensions

The techniques underpinning Shapley value distillation can be generalized:

• To distillation of other influence indices (such as Banzhaf), via analogous analytic and boosting frameworks.
• To distributed and federated learning, where models may need to be reconstructed from aggregate attributions rather than raw data.
• To “inverse explainability” scenarios in XAI, in which one infers or constrains models/classes of models from observed attributions, enforcing interpretability or compliance requirements.

Shapley value distillation as rigorously instantiated in (De et al., 2012) provides the algorithmic foundation for these enterprises, coupling game-theoretic attribution, harmonic analysis, and boosting-based estimation in a single, unified methodology.