Union Shapley Value in Cooperative Games

• Union Shapley Value is a group extension of the Shapley value that allocates Harsanyi dividends among overlapping coalitions.
• It is defined using potential and balanced contributions axioms, ensuring fairness through a clear marginal impact formulation.
• The framework supports analyses in economics, machine learning, network science, and multi-agent systems by quantifying collective impact.

The Union Shapley Value is a principled extension of the Shapley value from individual players to groups, quantifying a group's contribution to the overall performance of a cooperative game through the aggregated impact of the group's removal across all coalitions. This value is formally defined by collecting the Harsanyi dividends of all coalitions that overlap with the group, distributing each coalition's dividend equally among its members. The framework offers rigorous axiomatizations, unveils deep connections to previously proposed group-centric values, and introduces a dual theory focused on synergy quantification rather than aggregate worth. Methodologically, the Union Shapley Value provides a robust lens for analyzing collective impact and collaborative dynamics in cooperative games, with direct applications in economics, machine learning, network science, and multi-agent systems.

1. Definition and Mathematical Formulation

The Union Shapley Value (USV) extends the classic Shapley value from individuals to arbitrary groups of players. In a game $(N, v)$, for any group (coalition) $S \subseteq N$, the value is defined using the Harsanyi dividend, $\Delta_v(T)$, which quantifies the surplus generated by coalition $T$ beyond its strict subcoalitions. The formal definition is:

$$US_S(N, v) = \sum_{T \subseteq N : S \cap T \neq \emptyset} \frac{\Delta_v(T)}{|T|}$$

where each dividend is divided equally among members of $T$. An equivalent marginal-impact formulation is:

$$US_S(N, v) = \sum_{T \subseteq N} \frac{(|T|-1)! (|N|-|T|)!}{|N|!} \cdot [v(T) - v(T \setminus S)]$$

This captures the aggregated “removal impact” of the group $S$ over all coalitions it intersects, thereby measuring the collective importance of $S$ in $v$ (Kępczyński et al., 27 May 2025).

2. Axiomatic Characterizations

Two central axiomatizations establish the USV as the canonical group extension of the Shapley value.

a) Potential Axiom: There exists a potential function $P(\cdot, v)$ on games such that the USV of $S$ equals the drop in potential upon removing $S$:

$US_S(N, v) = P(N, v) - P(N \setminus S, v)$

with $$P(N, v) = \sum_{T \subseteq N, T \neq \emptyset} \frac{\Delta_v(T)}{|T|}$$.

b) Balanced Contributions Axiom: For any two coalitions $S, T \subseteq N$, the reciprocal removal impacts balance:

$$US_S(N, v) - US_{S\setminus T}(N\setminus T, v) = US_T(N, v) - US_{T\setminus S}(N\setminus S, v)$$

These axioms generalize the classic Shapley fairness and consistency axioms from individuals to arbitrary groups, ensuring that USV is the unique value satisfying these properties and reducing to the classic Shapley value on singletons (Kępczyński et al., 27 May 2025).

3. Group Semivalues: Weighted Aggregation Schemes

A broader family of group values, termed group semivalues (Editor's term), is characterized by weighted sums over Harsanyi dividends of coalitions intersecting the group:

$$\varphi_S(N, v) = \sum_{T \subseteq N : S \cap T \neq \emptyset} p^{|S \cap T|}_{|T|} \cdot \Delta_v(T)$$

where the coefficients $p^{q}_{t}$ depend only on the group-coalition size relationship. Special cases include:

• Union Shapley Value: $p^{q}_{t} = 1/t$ for all $q, t$.
• Sum of Individual Shapley Values: $p^{q}_{t} = q/t$.
• Merge Shapley Value: another specific weighting.

These constructs maintain linearity, symmetry, and consistency with the Shapley value on singletons, and provide a unified framework for interpreting various group-centric allocation rules in games (Kępczyński et al., 27 May 2025).

4. Synergy-Oriented Dual Extension

Distinct from USV, the theory introduces synergistic semivalues, focusing on interactions where the entire group $S$ is present. The intersection-based value (“Intersection Shapley Value”) is:

$$IS_S(N, v) = \sum_{T : S \subseteq T \subseteq N} \frac{\Delta_v(T)}{|T|}$$

This quantifies pure synergy: the value generated only when all members of $S$ act together. Key relationships include:

$$US_{\{i, j\}}(N, v) = SV_i(N, v) + SV_j(N, v) - IS_{\{i, j\}}(N, v)$$

and, more generally, the inclusion–exclusion formula:

$$US_S(N, v) = \sum_{\emptyset \neq T \subseteq S} (-1)^{|T|-1} IS_T(N, v)$$

This delineates aggregate group contributions from their synergistic overlap, providing precise quantification of redundancy and complementarity within groups (Kępczyński et al., 27 May 2025).

5. Connections to Existing Group Values

The USV framework exposes formal links among various previously studied group values:

• The union and intersection mechanisms are mutually dual under inclusion–exclusion.
• The interaction index (in literature) is dual to the Merge Shapley Value.
• The sum of Shapley values corresponds to a semivalue with $|S| \cdot IS_S$ weighting.

Such connections clarify the spectrum between additive, merged, and interactive group valuations, facilitating appropriate selection based on the analysis goal (e.g., total group impact vs. synergy measurement) (Kępczyński et al., 27 May 2025).

6. Applications and Implications

The Union Shapley Value and its semivalue relatives have broad relevance:

• Game Theory and Cost Allocation: Design of group-fair profit sharing and cost allocations that reflect both aggregate and interactive group effects.
• Machine Learning/Feature Attribution: Quantitative importance of feature groups (e.g., sets of neurons or predictors) underpinning interpretability in model explanations.
• Network Science/Risk Management: Quantification of the collective impact of node, agent, or risk factor removal.
• Multi-agent Systems: Assessment of collaborative team value under uncertainty or dynamic coalition formation.

Our results indicate that dual synergistic semivalues can enrich team formation, redundancy detection, and adaptive coalition analysis by distinguishing cooperative benefit from individual merit (Kępczyński et al., 27 May 2025).

7. Summary of Theoretical Contributions

The Union Shapley Value quantifies the group-level impact in cooperative games by equitably aggregating dividends over all coalitions that overlap the target group, underpinned by axiomatic characterizations analogous to those of the classical Shapley value. The framework unifies various group and synergy-centric allocation rules, provides explicit dualities, and extends the scope of fair value assignment from individual players to structured collectives. These advances facilitate principled and interpretable analysis of cooperative phenomena in both economic and computational domains.