---
title: 'Coalition Path View: Group Attribution'
url: https://www.emergentmind.com/topics/coalition-path-view
type: topic
---

# Coalition Path View: Group Attribution

The Coalition Path View describes a principled approach to quantifying the collective impact of a group—referred to as a coalition—in cooperative games, model explanation, and sensitivity analysis, with a focus on extensions of the Shapley value for groups. It provides a unifying perspective on group attributions, emphasizing the marginal effect of a group's collective presence or removal on the global value or outcome of interest. Central to this view is the Union Shapley value, a canonical group extension of the classical Shapley value justified via coherent axiomatic characterizations. The Coalition Path View clarifies the interplay between group value, group synergy, and the broader landscape of group semivalues and their computational properties, with direct implications for explainable AI, sensitivity analysis, and fairness allocation [2505.21122].

## 1. Foundations: Coalitional Games, Harsanyi Dividends, and Potential

A coalitional game is given by a tuple $(N, v)$, where $N$ is a finite player set and $v\colon 2^N \to \mathbb{R}$ is a set function satisfying $v(\emptyset)=0$. The Harsanyi dividend $\Delta_v(T)$ for $T\subseteq N$ is defined by the Möbius inversion
\[
\Delta_v(T) = \sum_{S\subseteq T}(-1)^{|T|-|S|} v(S),\qquad v(S)=\sum_{T\subseteq S} \Delta_v(T).
\]
The global “potential” of a game aggregates all coalitional effects:
\[
P(N, v) = \sum_{T\subseteq N,\, T\neq \emptyset} \frac{\Delta_v(T)}{|T|}.
\]
These structures support the definition of group-centric value functions and enable algebraic manipulations foundational to the Coalition Path View [2505.21122].

## 2. Union Shapley Value: Mathematical Definition and Structural Insights

The Union Shapley value $US_S(N, v)$ quantifies the marginal effect of the coalition $S\subseteq N$ by evaluating the drop in potential when $S$ is removed:
\[
US_S(N, v) = P(N, v) - P(N\setminus S, v) = \sum_{T\subseteq N\colon T\cap S\neq \emptyset} \frac{\Delta_v(T)}{|T|}.
\]
Alternatively, the path-based marginal contribution formula is
\[
US_S(N, v) = \sum_{T\subseteq N} \frac{(|T|-1)!\, (|N|-|T|)!}{|N|!} \big[v(T)-v(T\setminus S)\big].
\]
This expression cleanly generalizes the classical Shapley value (recovered for $|S|=1$) and formalizes the coalition path as the process of sequentially removing $S$ and observing the overall decrement in potential. The Union Shapley value operates as a “collective removal” metric, allocating to $S$ the credit for every coalition $T$ for which $T\cap S\ne\varnothing$, partitioning dividends appropriately [2505.21122].

## 3. Axiomatic Characterization and Semivalue Relationships

Two parallel axiomatic characterizations uniquely identify the Union Shapley value:
- **Potential + Shapley consistency:** There exists $f$ such that $\varphi_S(N, v) = f(N, v) - f(N\setminus S, v)$ and for $|S|=1$, $\varphi_S$ coincides with the Shapley value.
- **Balanced contributions + Shapley consistency:** For all $S, T\subseteq N$,
\[
\varphi_S(N, v) - \varphi_{S\setminus T}(N\setminus T, v) = \varphi_T(N, v) - \varphi_{T\setminus S}(N\setminus S, v),
\]
with $\varphi_\emptyset\equiv 0$.
Both axiomatizations replace the classical efficiency axiom with group-level or singleton-level modifications and retain group symmetry (linearity) and a null-player principle adapted to groups [2505.21122].

The Union Shapley value nests within the broader class of **group semivalues**:
\[
\varphi_S(N, v) = \sum_{T\subseteq N\colon T\cap S\neq\emptyset} p^{|S\cap T|}_{|T|} \Delta_v(T),
\]
where the Union Shapley choice is $p^q_t = 1/t$, but alternatives (e.g., Merge Shapley, sum of singleton Shapleys) use different $p$-families. The dual semivalue, induced by Möbius duality, is the Intersection Shapley value, which quantifies synergy:
\[
IS_S(N, v) = \sum_{T\supseteq S} \frac{\Delta_v(T)}{|T|}.
\]
Key inclusion-exclusion relations intertwine Union and Intersection Shapley values, reflecting a deep combinatorial symmetry [2505.21122].

## 4. Computational Tractability and Algorithmic Aspects

All group semivalues and their duals require summation over $2^n$ coalitions. No polynomial-time algorithm exists for general $n$, but Monte Carlo and random sampling strategies (as in classical Shapley approximation) are effective in practice. For Union Shapley, it suffices to sample coalitions $T$ with $T\cap S\ne\varnothing$ and aggregate $\Delta_v(T)/|T|$. For Intersection Shapley, sample $T\supseteq S$. The paper proposes future work in polynomial-time approximations but provides no such specialized algorithms [2505.21122].

## 5. Interpretive Examples: Distinguishing Group Impact and Synergy

The “12 ∣ 34” game, with $N=\{1,2,3,4\}$ and $v(S)$ equals $1$ if $S$ contains either $\{1,2\}$ or $\{3,4\}$, vividly illustrates the discriminative power of the Union Shapley value. Merge Shapley and the sum-of-singletons both assign value $1$ to $\{1,2\}$ and $\{1,3\}$, yet their removal has qualitatively different effects:
- $US_{\{1,2\}} = \frac{1}{2}$ (removal still leaves a winning coalition);
- $US_{\{1,3\}} = 1$ (removal destroys all positive coalitions).
The Intersection Shapley value dually quantifies synergy:
- $IS_{\{1,2\}} = \frac{1}{2}$,
- $IS_{\{1,3\}} = 0$.
These metrics allow clear separation of group “worth” (collective impact) and group “synergy” (non-additive cooperative effects), equipping the Coalition Path View with operational expressiveness not available to singleton-based indices [2505.21122].

## 6. Theoretical and Practical Significance

The Coalition Path View, instantiated by the Union Shapley value, provides a conceptually rigorous, axiomatically unique, and algebraically tractable paradigm for quantifying group-level contributions in coalitional scenarios. It clarifies the allocation of shared value, disentangles group effects from mere summation of individuals, and connects classical results on interaction indices and semivalue theory. The explicit duality between removal (Union) and synergy (Intersection) values further anchors the theory within the broader landscape of cooperative game theory, sensitivity analysis, and modern explainable AI paradigms. The challenge of exponential computational scaling is mitigated partially by sampling and approximation strategies, with algorithmic efficiency remaining a prominent research direction [2505.21122].

## 7. Connections and Implications

The Coalition Path View sharply differentiates itself from naïve grouping (e.g., sum of individual Shapleys or pure conditional worth) by enforcing fair, balanced attributions via Möbius-weighted summation across the whole power set of coalitions. Its formal structure supports extensions to fairness, feature selection, significance testing, and synergistic group analysis in machine learning and decision science. The path-based removal intuition, together with the axiomatic underpinnings, suggests applicability far beyond traditional cooperative games, aligning with recent trends in model interpretability and attribution for complex, structured data domains [2505.21122].

Source: https://www.emergentmind.com/topics/coalition-path-view