Synergistic Semivalues in Information and Game Theory

• Synergistic semivalues are formal constructs in information and game theory that quantify the extra value from collective interactions.
• They implement the 'whole minus union' principle by isolating genuine collective benefits from redundant and individual contributions.
• Their applications in machine learning, neuroscience, and network analysis enable precise attribution of higher-order interactive effects.

Synergistic semivalues are formal constructs, originating within information theory and cooperative game theory, that assign a quantitative "value" specifically to the synergistic contribution arising when elements act in coalition—beyond what is attributable to the union, sum, or norm of their individual effects. In the context of multivariate information, synergistic semivalues quantify the informational gain achieved only through the joint, cooperative interaction of predictors, while in game theory they capture the "extra" or "dual" payoff generated collectively above additive individual or pairwise contributions.

1. Information-Theoretic Basis: The Whole Minus Union Principle

A foundation for synergistic semivalues is the notion that synergy is the difference between the information a set of variables provides collectively about a target and the information obtainable from the union of their individual contributions. This is formalized by a measure of the form: $$S_{\text{vkk}}(\{X_1, \ldots, X_n\} : Y) = I(\{X_1,\ldots, X_n\} : Y) - I_{\text{vkk}}(\{X_1,\ldots, X_n\} : Y)$$ where $I_{\text{vkk}}$ ("union information") is computed by minimizing joint mutual information over distributions with fixed marginals: $$I_{\text{vkk}}(\{X_1,\ldots, X_n\} : Y) = \min_{p^*} I^*(X_1, ..., X_n : Y)$$ subject to $p^*(X_i, Y) = p(X_i, Y)$ for each $i$. Here, $I^*$ is the mutual information under the "noisified" joint distribution, effectively removing higher-order dependencies while preserving marginals. This approach isolates the surplus, truly cooperative information, removing both redundancy and unique information (Griffith et al., 2012).

This "whole minus union" paradigm aligns with key axiomatic desiderata: symmetry under permutation, self-redundancy, monotonicity, and, where possible, positive definiteness.

2. Synergistic Semivalues in Cooperative Game Theory

The parallel to cooperative game theory is explicit. Classical semivalues, as generalizations of the Shapley value, distribute total game worth among players according to marginal or cooperative contributions. The conceptual analogy for synergy is:

• Consider the "value" of a coalition as the function $v(S)$.
• Define the "union value" as the aggregate of individual or overlapping contributions.
• The synergistic semivalue for a coalition S is then $v(S)$ minus its union value, reflecting the surplus only available to the group as a whole.

This link has been formalized through the development of group semivalues, including the Union Shapley value and dual constructs such as the Intersection Shapley value (IS), which aims to capture only the irreducible synergy attributable to a coalition S (Kępczyński et al., 27 May 2025). The IS is constructed as: $$\text{IS}_S(N, v) = \sum_{T \supseteq S} \frac{\Delta v(T)}{|T|}$$ where $\Delta v(T)$ denotes the Harsanyi dividend for coalition T. The IS thus quantifies the excess that only arises if all members of S are present, aligning with the definition of pure synergy.

3. Axiomatic and Dual Characterizations

The Union Shapley value (US), as a group extension, is defined axiomatically via potential drop or balanced contributions, and has explicit duality with the Intersection Shapley value. For any group S: $$\text{US}_S = \text{Sum of individual SVs} - \text{IS}_S$$ This decomposition demonstrates that the synergistic semivalue (IS) represents that portion of group value double-counted (as overlap) in the naive sum of individual Shapley values and should therefore be subtracted to avoid redundancy.

A table summarizing key forms follows:

Name	Formula	Interpretation
Union Shapley (US)	$$\sum_{T: S \cap T \neq \emptyset} \frac{\Delta v(T)}{|T|}$$	Value lost by removing group S
Intersection Shapley (IS)	$\sum_{T \supseteq S} \frac{\Delta v(T)}{|T|}$	Pure synergy of coalition S
Synergy (IS for pair i,j)	$SV_i + SV_j - US_{\{i,j\}}$	Excess beyond sum of individuals

This framework reflects the more general semivalue structure: $$\phi_S(N, v) = \sum_{T: S \cap T \neq \emptyset} p_{|T|}^{|S \cap T|} \Delta v(T)$$ where choice of weights $p_{|T|}^{|S \cap T|}$ adjusts the value for specific synergistic or aggregate perspectives (Kępczyński et al., 27 May 2025).

4. Partial Information Decomposition and Synergistic Information

In information theory, partial information decomposition (PID) provides fine-grained attribution of mutual information into unique, redundant, and synergistic pieces. Synergistic semivalues here quantify information about the target obtainable only from combinations, not from any source alone. For example, in Gaussian systems, the MMI PID assigns redundancy as the minimum mutual information and synergy as the surplus above individual source contributions: $$S(X;Y,Z) = I(X;Y,Z) - I(X;Y) - I(X;Z) + \min\{I(X;Y), I(X;Z)\}$$ (Barrett, 2014). In more general settings, various approaches (including constrained minimization, geometric pooling, or orthogonal decompositions) operationalize the "whole minus union" principle for synergy values.

Synergistic semivalues, in this context, must be nonnegative, symmetric, and robust to duplication of predictors, and often emerge as the positive part of the difference between total mutual information and union or pooled information (Gomes et al., 25 Mar 2024, Enk, 2023).

5. Pooling, Lattice Structure, and Generalizations

Recent developments interpret the "union" operation for information not simply as an arithmetic sum but as the output of an optimal pooling of probability distributions associated with each part. The form: $$I_{\text{syn}} = H_{\text{pooled}} - H_{\text{joint}}$$ relies on the pooling method: geometric averaging, conditional-independence construction, or other channel-based schemes (Enk, 2023, Gomes et al., 25 Mar 2024). Each node in the corresponding information lattice (distinct from the redundancy-based PID lattice) is associated with a specific pooled distribution and thus with its own entropy "value."

This approach renders synergistic semivalues as differences in uncertainty between the pooled and the joint distribution—a direct analog of the dividend-driven semivalues in game theory.

6. Applications and Interpretative Significance

Synergistic semivalues provide key operational tools in multiple domains:

• In feature attribution for machine learning, they distinguish between synergistic, redundant, and independent feature interactions, enabling rigorous credit assignment beyond Shapley or SHAP values (Ittner et al., 2021).
• In neuroscience and complex systems, synergistic semivalues allow the quantification of joint information processing beyond summing parts, revealing integration or distributed computation (Barrett, 2014, Quax et al., 2016).
• In network analysis and fairness studies, synergistic group values guide intervention, connectivity, and robust allocation by capturing irreducible collective impact.

By unifying information-theoretic, probabilistic, and game-theoretic measures, synergistic semivalues constitute a rigorous backbone for analyzing multi-agent or multi-variable systems in which higher-order dependencies drive the main functional properties.

7. Open Questions and Future Directions

Open problems include:

• Extending the robust axiomatization and computability of synergistic semivalues to settings with more than two predictors or large coalitions, especially where redundancy and synergy are not cleanly separable.
• Developing practical algorithms for estimating such values in high-dimensional real data, with attention to finite-sample effects and model misspecification (Takimoto, 2020).
• Integrating synergistic semivalues into real-world decision-making, such as coalition formation, feature selection, and the design of interpretable or fair AI systems.

The entwined evolution of semivalue theory in both cooperative game theory and multivariate information science suggests a broad and deepening impact, with synergistic semivalues serving as the precise quantifiers of collective value in increasingly complex data-driven domains.