Intersection Shapley Value

• Intersection Shapley Value is a generalized solution concept that allocates surplus fairly in overlapping coalition structures using intersection-closed systems.
• It employs a uniform-dividend approach by solving linear systems to ensure efficiency, additivity, and equal treatment under restricted axioms.
• Its applications span explainable machine learning, network analysis, and federated systems, offering robust allocations in settings with incomplete coalition data.

The Intersection Shapley Value is a generalized solution concept in cooperative game theory that addresses value allocation in settings characterized by overlapping coalitional structures. It is particularly relevant for games in which the set system of known coalition values is intersection-closed—meaning that the intersection of any two known coalitions remains known. The concept finds utility in incomplete cooperative games, machine learning contexts involving interactions among features, and network scenarios with overlapping group memberships, by rigorously extending the Shapley value’s axiomatic fairness to allocations over intersecting coalitional families.

1. Formal Definition in Intersection-Closed Systems

The Intersection Shapley Value, as articulated through the uniform-dividend value (UD-value) (Černý, 9 Jan 2025), is defined for cooperative games with an incomplete characteristic function: only the worths of a subset $K$ of coalitions are known a priori. The intersection-closed property enforces that for any $S, T \in K$, their intersection $S \cap T$ also lies in $K$. The closure mapping $c_K(T) = \bigcap \{S \in K : T \subseteq S\}$ identifies families of "indistinguishable" coalitions.

For each subset $S \subseteq N$ of players, the UD-value determines the dividends $\delta_v^K(S)$ by solving:

• For every $S \in K$, $\sum_{T \subseteq S} \delta_v^K(T) = v(S)$
• For $S, T$ with $c_K(S) = c_K(T)$, $\delta_v^K(S) = \delta_v^K(T)$

Given these dividends, the allocation to player $i$ is

$$\Phi^{(K)}_i(v) = \sum_{\substack{S \subseteq N \ i \in S}} \frac{\delta_v^K(S)}{|S|}$$

This construction generalizes the classical Shapley value to incomplete games by distributing surplus uniformly among all indistinguishable coalitions, ensuring allocations respect observed coalition structure.

2. Key Properties and Axiomatic Characterization

The UD-value possesses a set of properties refined from the standard Shapley axioms, adapted for intersection-closed systems and incomplete games. These include:

• Efficiency: $\sum_{i \in N} \Phi^{(K)}_i(v) = v(N)$, guaranteeing the entire surplus is allocated.
• Additivity: For characteristic functions $v$, $w$, it holds that $$\Phi^{(K)}_i(v + w) = \Phi^{(K)}_i(v) + \Phi^{(K)}_i(w)$$.
• Equal treatment (restricted): Equal allocation to players in indistinguishable positions under the closure mapping.

Crucially, the null player and equal treatment axioms are enforced only for games within the restricted dividend space $\mathcal{UD}^K_n$, i.e., games where non-distinguishable coalitions must have identical dividends. These axiomatic requirements guarantee the UD-value is uniquely determined in intersection-closed settings (Černý, 9 Jan 2025).

3. Comparison to Related Allocation Rules

For intersection-closed games, alternative allocation rules are available:

• R-value: Shapley value applied to a pessimistically completed game with the surplus of unknown coalitions set to zero via recursive surplus assignment.
• IC-value: Shapley value applied to a monotonic completion defined by $v_{IC}(S) = v(c_K(S))$ for all $S \subseteq N$; this value is typically more egalitarian, distributing surplus evenly.

Empirically, the UD-value and the R-value yield similar allocations—demonstrated by smaller $\ell_1$-norm differences—while the IC-value is consistently closer to equal division and enforces monotonicity more strictly. The UD-value’s averaging over positive extensions provides a balance between robustness and sensitivity to marginal contributions (Černý, 9 Jan 2025).

Value	Completion Method	Fairness Profile
UD-value	Uniform dividend splits	Marginal, robust average
R-value	Zero unknown surpluses	Pessimistic, marginal
IC-value	Closure-based monotonic	Egalitarian, monotonic

4. Mathematical Structure and Practical Calculation

The dividend calculation is performed by solving a linear system:

• For each $S \in K$, enforce $\sum_{T \subseteq S} \delta_v^K(T) = v(S)$
• Set $\delta_v^K(S) = \delta_v^K(T)$ for closures $c_K(S) = c_K(T)$

Allocations are then summed (across all coalitions containing $i$, normalized by coalition size), in analogy to the classical Shapley formula. For the R-value, the surplus for unknown coalitions is always zero; for the IC-value, coalition values are inflated via closure assignment.

Simulations over systems with $3 \leq n \leq 6$ players reveal increasing uniqueness of the UD-value for non-intersection-closed systems as $n$ grows. This indicates the UD-value’s practical viability as an allocation rule even beyond strictly intersection-closed settings.

5. Broader Implications and Extensions

The uniform-dividend interpretation of the Intersection Shapley Value as the expected Shapley value over all positive (nonnegative-surplus) game extensions provides a principled mechanism for allocating surplus under uncertainty (Černý, 9 Jan 2025). This approach is particularly suited for applications in which incomplete information about coalition worths is the rule rather than the exception, such as federated learning (overlapping data sources), explainable AI (feature interactions), or distributed systems.

Empirically, the IC-value’s tendency toward more egalitarian splits may be less suitable when marginal contributions vary significantly, while the UD-value’s sensitivity to observed dividends produces robust allocations that remain consistent across a wide range of set systems. A plausible implication is that the UD-value can be seen as a specialization of the Intersection Shapley Value for intersection-closed systems but also as a general allocation rule for incomplete games with substantial information gaps.

6. Connections to Interaction Indices and Overlapping Group Structures

More broadly, allocation rules based on intersections are intimately connected with recent work in explainable machine learning, such as the Faith-Shap interaction index (Tsai et al., 2022) and feature selection settings (Fryer et al., 2021, Rozemberczki et al., 2022). In those contexts, the allocation must also consider interactions among overlapping feature groups or agent coalitions. Uniform-dividend principles, closure mappings, and dividend equality constraints are recurring motifs in such settings where intersection structure dictates allocation fairness.

Recent work suggests formulating intersection-based Shapley values via restriction of permissible coalition permutations or via averaging over admissible game extensions, paralleling the UD-value’s methodology. This makes the uniform-dividend value relevant beyond classical cooperative game theory, with potential applications in rigorous data and feature valuation, network centrality under hypergraph structure (Yamada, 2021), and stochastic path integral extensions (Lim, 2022) for coalition formation.

7. Conclusions

The Intersection Shapley Value, exemplified by the UD-value for intersection-closed set systems, is a rigorous and principled allocation rule capturing fair dividends where coalition worths are partially known and group memberships overlap. Its unique specification by closure-based dividends, efficiency, and restricted axiomatic constraints make it computationally tractable, empirically robust, and theoretically grounded as an expected Shapley value over positive game completions. Comparison with R-value and IC-value demonstrates nuanced trade-offs between fairness and marginality, while its structure suggests natural generalizations for incomplete game settings and interacting systems.

The growing fraction of non-intersection-closed systems yielding unique UD-value allocations as player count increases further highlights its practical appeal. When surplus information is incomplete and coalition overlaps are prevalent, the Intersection Shapley Value provides a canonical solution for equitable and robust value allocation.