Shapley Value Decomposition Methods

• Shapley Value-Based Decomposition Methods are approaches that use decomposability in cooperative games to fairly distribute value based on players’ marginal contributions.
• They enhance computational efficiency by breaking down high-dimensional problems into independent subgames, reducing exponential complexity to manageable polynomial tasks.
• These methods find practical applications in multiterminal data compression, portfolio attribution, and resource allocation, offering both theoretical robustness and actionable workflows.

Shapley Value-Based Decomposition Methods

The Shapley value, a fundamental solution concept in cooperative game theory, provides an axiomatic framework for fairly allocating the total value (or cost) of a coalition among players, based on their marginal contributions. In complex, high-dimensional, or structured systems, direct Shapley value computation is often intractable due to the combinatorial explosion in the number of coalitions. Shapley value-based decomposition methods systematically exploit structure—such as independence, decomposability, and submodularity—to enable scalable, principled, and interpretable allocations in settings ranging from multiterminal data compression and portfolio attribution to networked systems and allocation games.

1. Classical Shapley Value and Decomposition Theorems

The classical Shapley value for a cooperative game $(N, v)$ with $n=|N|$ players is defined as

$$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$

For convex games with cost functions satisfying submodularity, the core is nonempty and the Shapley value lies uniquely in the core.

A central result for structured games is the Shapley Decomposition Theorem: if the characteristic function $v$ (e.g., a submodular cost such as entropy) is decomposable, i.e., there exists a partition $P=\{C_1,\ldots,C_k\}$ of $N$ such that

$v(N) = \sum_{C\in P} v(C),$

and $v(C\cup C')=v(C)+v(C')$ for $C\cap C'=\emptyset$, then both the core and the Shapley value factorize:

• Core: $C(N) = \bigoplus_{C\in P} C(C)$,
• Shapley value: $n=|N|$0 (Ding et al., 2018).

Thus, $n=|N|$1 depends only on the subgame in its partition block $n=|N|$2; the total computation reduces to smaller, independent subproblems.

2. Algorithmic Decomposition and Complexity

The decomposition approach operates as follows:

1. Find the finest decomposer: Identify the partition $n=|N|$3, the maximal decomposition into independent blocks, via submodular minimization (e.g., minimizing $n=|N|$4 over all $n=|N|$5 for entropy games); see Algorithm 1 in (Ding et al., 2018).
2. Compute subgame Shapley values: For each $n=|N|$6, compute $n=|N|$7 via the classical formula, restricted to $n=|N|$8.
3. Aggregate global Shapley value: Concatenate subgame values into the global vector: $n=|N|$9.

The computational complexity improves dramatically:

• Direct: $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$0, where $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$1 is the cost of a function evaluation.
• Decomposed: After $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$2 for partition finding, each block $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$3 requires $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$4. Maximum block size $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$5 yields total complexity $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$6, a substantial reduction for modest $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$7 (Ding et al., 2018).

For sufficiently decomposable games, the bottleneck shifts from exponential to polynomial in $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$8 for block enumeration, enabling tractable computations for large systems.

3. Applications: Data Compression, Portfolio Attribution, Allocation

Multiterminal Data Compression

In multiterminal Slepian-Wolf coding, sources (sensors) are modeled as players, and the joint entropy function provides the submodular cost. The decomposability of inter-source dependency (e.g., statistical independence between subsets) enables the decomposition method to efficiently yield a fair rate-allocation vector in the Slepian-Wolf region, consistent with both information-theoretic optimality and Shapley fairness (Ding et al., 2018).

Performance Attribution in Finance

Shapley decomposition attributes portfolio performance to individual features or investment signals, considering all possible interactions (Moehle et al., 2021). Exact enumeration is feasible only for small numbers of features, but sampling and structural decomposition—where certain sets of features operate independently—permit accurate and interpretable attributions in complex strategies.

Allocation Games

In resource allocation with indivisible goods, decomposability can arise when agents' preferences or goods split into disjoint clusters; each cluster can be solved independently (Lupia et al., 2017). Algorithmic decompositions, coupled with approximation schemes (FPRAS, range-based sampling), yield tight bounds and efficient computations at scale.

4. Structural and Convex-Analytic Foundations

Shapley decomposition is rooted in the theory of submodular base polyhedra and convex games. When the characteristic function is submodular (e.g., entropy, cut functions), the base polyhedron $$\phi_i(v) = \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!\,(n-|S|-1)!}{n!} [v(S\cup\{i\})-v(S)].$$9 coincides with the game core. Its extreme points, corresponding to orderings of the ground set, structurally factor along decomposable partitions: $v$0 The Shapley value is the barycenter of these extreme points, thus inherits the block-wise separability (Ding et al., 2018).

Moreover, for any submodular cost game, the decomposition theorem applies directly:

• Network design: cut cost games decompose by disconnected components.
• Facility location: independent facilities induce block structure.

5. Extensions, Generalizations, and Limitations

Beyond Entropy: General Submodular Games

Any cooperative game with submodular characteristic function admits decomposition as above (Ding et al., 2018). This includes myriad combinatorial optimization problems where costs or rewards decompose due to independence, conditional independence, or modularity.

Parallel Computation

Decomposed subgames can be solved in parallel, reducing wall-clock time from exponential to $v$1, with $v$2 the maximal block size, further enhancing scalability in distributed environments (Ding et al., 2018).

Limitations

The method requires exact or approximate decomposability, and its efficiency gain is directly tied to the maximal block size after partitioning. If interactions are dense (i.e., all components interact), the decomposition collapses back to the full exponential problem.

Approximate versions—sampling or hybridizations—can address modest violations of strict decomposability, but do not guarantee exact Shapley allocations.

6. Illustrative Example and Practical Workflow

Consider sources $v$3 where $v$4 and $v$5 are correlated, $v$6 is independent:

• $v$7.
• Compute Shapley value for block $v$8 (two-player formula), and assign $v$9 to player $P=\{C_1,\ldots,C_k\}$0.
• The global Shapley vector is the direct sum: $P=\{C_1,\ldots,C_k\}$1 (Ding et al., 2018).

For general workflow:

1. Input game function (e.g., entropy, cost, utility).
2. Detect decomposable partition via submodular minimization.
3. Compute subgame Shapley values.
4. Aggregate.

This pipeline generalizes across domains—network design, economics, operations research—leveraging structure for scalable, fair allocation.

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References

• "Fairness in Multiterminal Data Compression: Decomposition of Shapley Value" (Ding et al., 2018)
• "Portfolio Performance Attribution via Shapley Value" (Moehle et al., 2021)
• "Computing the Shapley Value in Allocation Problems: Approximations and Bounds..." (Lupia et al., 2017)