Beyond Shapley Values: Cooperative Games for the Interpretation of Machine Learning Models (2506.13900v1)

Abstract: Cooperative game theory has become a cornerstone of post-hoc interpretability in machine learning, largely through the use of Shapley values. Yet, despite their widespread adoption, Shapley-based methods often rest on axiomatic justifications whose relevance to feature attribution remains debatable. In this paper, we revisit cooperative game theory from an interpretability perspective and argue for a broader and more principled use of its tools. We highlight two general families of efficient allocations, the Weber and Harsanyi sets, that extend beyond Shapley values and offer richer interpretative flexibility. We present an accessible overview of these allocation schemes, clarify the distinction between value functions and aggregation rules, and introduce a three-step blueprint for constructing reliable and theoretically-grounded feature attributions. Our goal is to move beyond fixed axioms and provide the XAI community with a coherent framework to design attribution methods that are both meaningful and robust to shifting methodological trends.

Cooperative game theory provides a powerful framework for interpreting machine learning models by treating features as players contributing to a prediction or other model output. While Shapley values have become the dominant method in this space, particularly through techniques like SHAP [Lundberg2017], this paper argues for exploring a broader range of tools from cooperative game theory to develop more robust and meaningful feature attributions. The authors highlight that the reliance on axiomatic justifications for Shapley values in XAI contexts can be problematic, as the relevance of these axioms to feature importance is debatable.

The paper introduces a more general perspective on cooperative games for machine learning interpretability, focusing on the concept of efficient allocations. An allocation assigns a value or "payoff" to each feature such that the sum of these values equals the total "value" produced by the full set of features (minus the value of no features, $v(D) - v(\emptyset)$). This efficiency property ensures that the quantity being explained (e.g., the model's prediction $f(x)$) is fully decomposed among the features.

Two key families of efficient allocations are presented as alternatives and generalizations of Shapley values: the Weber set and the Harsanyi set.

1. The Weber Set: This set of allocations is based on the idea of features joining a coalition in a random order. Each feature's contribution in a specific ordering is its marginal contribution when joining the features already present in that order. An allocation from the Weber set is the expected marginal contribution averaged over all possible orderings, according to a probability distribution over permutations $p$. The Shapley value is a specific instance of a Weber set allocation where the probability distribution over permutations is uniform ($p(\pi) = 1/d!$ for all permutations $\pi$). Implementing a Weber set allocation involves defining the distribution $p$ and computing the expected marginal contribution. For a model $f$ and value function $v$, the allocation for feature $j$ is:

   $$\phi_v(j) = \mathbb{E}_{p} [v(\pi^j) - v(\pi^j \setminus \{j\})],$$

   where $\pi^j$ is the set of features preceding or including $j$ in permutation $\pi$. Computing this exactly requires summing over all $d!$ permutations, which is computationally prohibitive for large $d$. Practical implementations typically approximate this by sampling permutations. Tailoring allocations via the Weber set means choosing a non-uniform distribution $p$, which can incorporate prior knowledge, such as causal relationships between features [Frye2020].

2. The Harsanyi Set: This family of allocations is based on distributing the "dividends" of coalitions. Harsanyi dividends $\varphi_v(A)$ represent the unique value generated by a coalition $A$ that is not already accounted for by its sub-coalitions. The Harsanyi set allocations distribute these dividends among the members of each coalition according to a weight system $\lambda_j(A)$. The Shapley value is recovered when every dividend $\varphi_v(A)$ is divided equally among the $|A|$ members of coalition $A$. An allocation from the Harsanyi set for feature $j$ is:

   $$\phi_v(j) = \sum_{A \in \mathcal{P}_D : j \in A} \lambda_j(A) \varphi_v(A),$$

   where $$\varphi_v(A) = \sum_{B \subseteq A} (-1)^{|A|-|B|}v(B)$$. Computing Harsanyi dividends and then applying a weight system requires evaluating the value function $v$ for all $2^d$ subsets, similar to the challenge with standard Shapley value calculation methods based on marginal contributions. This summation can be approximated by sampling coalitions. The flexibility here lies in defining the weight system $\lambda$, allowing for custom rules on how coalition contributions are shared.

The authors argue that for practical XAI, the critical property is the efficiency of the allocation, ensuring the output is fully decomposed. They propose a three-step blueprint for developing robust game-theoretic feature attributions:

1. Choose a Quantity of Interest: Define precisely what needs to be explained. This could be a local quantity for a specific instance (e.g., $f(x)$) or a global property (e.g., variance of $f(X)$). Examples include model predictions, prediction uncertainty [Watson2023], or sensitivity measures [Iooss2019, Herin2024].
2. Choose a Value Function: Select a function $v:\mathcal{P}_D \rightarrow \mathbb{R}$ such that $v(D)$ equals the chosen quantity of interest. The selection of the value function is paramount, as it determines the underlying model representation being explained. Different value functions can satisfy $v(D) = \text{quantity}$ but lead to different interpretations. For prediction decomposition, common value functions involve conditional expectations $v(A) = \mathbb{E}[f(X) \mid X_A = x_A]$. However, for models with correlated features, these functions based on orthogonal projections can yield "impure" attributions that don't faithfully reflect the model's structure. Recent theoretical work suggests using value functions based on oblique projections [IlIdrissi2025] to address this, leading to more faithful decompositions.
3. Choose an Allocation: Select an efficient allocation scheme (like Shapley, or one from the Weber or Harsanyi sets) to redistribute the value $v(D)-v(\emptyset)$ among the features. While less critical than the value function, the allocation choice can highlight different aspects of feature contribution. For example, Proportional Marginal Effects (PME), an allocation adapted for variance decomposition tasks [Herin2024], can better identify spurious features compared to standard Shapley values when features are correlated.

Implementing this blueprint requires careful consideration of computational costs. Calculating exact value functions involving expectations often requires integration, which can be complex depending on the model and feature distributions. Approximating these values and the subsequent allocation (via sampling permutations or subsets) is common practice but introduces approximation errors. Libraries like SHAP provide implementations for various value functions and the Shapley allocation, often using approximation techniques optimized for specific model types. Extending these libraries to support custom value functions or alternative allocations from the Weber or Harsanyi sets would be a practical step towards leveraging this broader framework.

In conclusion, the paper advocates for a more principled and flexible use of cooperative game theory in XAI. By focusing on the efficiency property and separating the concerns of defining the quantity of interest, selecting an appropriate value function, and choosing an allocation, practitioners can design attribution methods better tailored to specific interpretability goals and potentially more robust to challenges like feature correlation. The Weber and Harsanyi sets provide the theoretical foundation for developing these novel allocation schemes beyond the standard Shapley values.