Shapley values and axioms¶
The Shapley value is the classical fair-division solution for cooperative games. In explainability it underpins SHAP-style feature attributions when coalition payoffs are aggregated with Shapley weights. This page states the definition, the four defining axioms, and the uniqueness result from Shapley (1953).
For Banzhaf indices and estimand comparison in ML pipelines, see Estimands: Shapley vs Banzhaf. For business deployment contexts, see Business and compliance applications.
Definition¶
For a cooperative game with player set \(N\), \(|N| = n\), and characteristic function \(v\), the Shapley value \(\phi_i(v)\) for player \(i \in N\) is:
Interpretation: imagine players join a coalition in random order, uniformly over all \(n!\) permutations. The Shapley value is player \(i\)’s expected marginal contribution when joining.
For coalition size \(k = |S|\) (excluding \(i\)), the weight is:
Implemented in ShapleyAggregator.
The four axioms¶
Shapley (1953) characterized \(\phi\) as the unique allocation rule satisfying four axioms on the space of cooperative games. Let \(\phi(v) = (\phi_1(v), \dots, \phi_n(v))\) denote the allocation vector for game \(v\).
Efficiency¶
The entire worth of the grand coalition is distributed. For games normalized so that \(v(\emptyset) = 0\):
Payoffs in mllm-shap are similarities to the base generation, so the empty
prompt usually scores above zero. The library therefore uses the convention
which is the axiom applied to the normalized game
\(v'(S) = v(S) - v(\emptyset)\): subtracting a constant from every payoff
leaves every marginal contribution, and hence every Shapley value, unchanged.
Each result reports both terms as
grand_value (\(v(N)\)) and
base_value (\(v(\emptyset)\)).
Explainability reading: attributions sum to the change in model utility
between the full prompt and the fully masked prompt. On the full coalition table
(exact, or any estimator at budget.fraction: 1.0) the identity holds to
floating-point precision and is the first consistency check in an audit; sampled
estimates approach it as the budget grows.
Symmetry¶
If players \(i\) and \(j\) make identical marginal contributions in every coalition — that is, \(v(S \cup \{i\}) = v(S \cup \{j\})\) for all \(S \subseteq N \setminus \{i, j\}\) — then:
Explainability reading: interchangeable tokens (e.g. duplicate padding) receive equal credit.
Dummy player (null player)¶
If player \(i\) never changes the payoff — formally \(v(S \cup \{i\}) = v(S)\) for all \(S \subseteq N \setminus \{i\}\) — then:
Explainability reading: tokens that do not affect the scored output receive zero attribution.
Additivity¶
For two games \(v\) and \(w\) on the same player set, define \((v + w)(S) = v(S) + w(S)\). Then:
Explainability reading: attributions decompose additively when utilities are sums of independent scoring components (for example, multi-objective value functions).
Uniqueness theorem¶
Theorem (Shapley, 1953). There exists exactly one allocation rule \(\phi\) that satisfies efficiency, symmetry, the dummy-player axiom, and additivity. It is given by the formula above.
This uniqueness result is why SHAP (Lundberg & Lee, 2017) argues for Shapley-based additive feature attributions under local accuracy and missingness constraints: within the class of additive explanation models, the Shapley kernel is the unique choice satisfying a parallel set of desirable properties.
Connection to SHAP in machine learning¶
Lundberg & Lee (2017) cast feature attribution as a cooperative game where features are players and \(v(S)\) is the model output (or a transform) when only feature subset \(S\) is observed. Under local accuracy (attributions sum to the prediction difference from a baseline) and missingness (absent features do not contribute), the Shapley value yields the SHAP explanation.
Important distinction for mllm-shap:
SHAP names a family of explanation methods tied to Shapley allocations.
An estimand is the quantity your pipeline actually estimates. Averaging the marginal contributions of sampled coalitions uniformly targets the Banzhaf index; averaging them within each coalition size first and then across sizes (a size-stratified mean) targets the Shapley value. The
mllm-shapaggregators do the latter for Shapley and the former for Banzhaf — see Estimands: Shapley vs Banzhaf and Approximate estimation.
When Shapley values are appropriate¶
Shapley allocations are a strong default when you need:
Efficiency for reconciliation with total model utility
Fair treatment of symmetric players
Additive decomposition across independent utility components
Regulatory narratives that reference the standard game-theoretic fairness axioms
They are not always the right estimand when you want pivotal-player (voting-power) interpretations; Banzhaf indices emphasize how often a player swings a coalition from losing to winning. See Estimands: Shapley vs Banzhaf for when the two diverge.
References¶
Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the Theory of Games II (pp. 307–317). Princeton University Press. DOI:10.1515/9781400881970-018
Shapley, L. S. (1952). A value for n-person games (RAND Paper P-295). RAND Corporation. RAND P-295
Lundberg, S. M., & Lee, S.-I. (2017). A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, 30. ML Anthology; arXiv:1705.07874
Roth, A. E. (Ed.). (1988). The Shapley Value: Essays in Honor of Lloyd S. Shapley. Cambridge University Press.
Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14(2), 65–72. DOI:10.1007/BF01769885