Shapley Value Allocation: Methods & Applications

• Shapley value allocation is a cooperative game theory framework that fairly divides the total coalition value by averaging marginal contributions based on symmetry, dummy, additivity, and efficiency axioms.
• It is widely applied in domains such as energy networks, online marketplaces, and supply chains to allocate costs, resources, and benefits efficiently.
• Advanced computational methods, including Monte Carlo sampling and clustering techniques, ensure scalable and accurate approximations in large systems.

Shapley value allocation is a foundational framework in cooperative game theory that allocates the total value generated by coalition-forming agents according to a principled fairness criterion based on marginal contributions. It is the unique solution satisfying symmetry, efficiency, additivity, and dummy player axioms, and has become the standard for resource, cost, and benefit allocation across domains such as energy networks, online marketplaces, finance, transport, and content platforms.

1. Definition and Fundamental Formula

Given a cooperative game (N, ν) where N = {1,…,n} is a finite set of players (agents, participants, or entities), and ν:2^N→ℝ is a characteristic function assigning a value to every coalition S⊆N, the Shapley value allocates to each player i the average of its marginal contributions across all coalition orders:

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

This formula evaluates all possible permutations (orders) of the players and averages the benefit brought by player i upon joining each possible existing coalition S. Key properties satisfied by the Shapley value are:

• Symmetry: Identical contributors receive identical payoffs
• Dummy: If a player never adds value, its allocation is zero
• Additivity: Shapley values are linear in the valuation function
• Efficiency: Allocations sum to ν(N), the total value of the grand coalition

2. Representative Applications in Large-Scale Resource and Profit Allocation

Energy and Network Systems

In distribution networks, Shapley allocation is used to share the cost of infrastructure upgrades and operation among energy consumers or energy communities. The characteristic cost function v(S) models long-run marginal cost or cost-saving for coalition S, often using stochastic or optimization-based simulation of network operation. For example, in the Turvey-Shapley method for grid cost allocation, v(S) is defined by probabilistic LRMC (long-run marginal cost) computed under load uncertainty, and clustering is used to make Shapley-value computation tractable for large N, yielding allocations that correlate strongly with coincident peak indicators and outperform naive energy-based rules in fairness and cost-causality (Azuatalam et al., 2019).

In multi-community energy systems engaging with distribution locational marginal prices (DLMPs), bilevel optimization incorporates DER operation and network constraints, with Shapley value applied to the characteristic function v(S) equals system cost savings for coalition S. Computational scalability is achieved via “signature-based” grouping of coalitions to minimize the number of required MILP solves, ensuring ex-post allocations retain Shapley axiomatic fairness (Park et al., 25 Oct 2025).

Crowd-Sourced and Platform Economies

For crowd-sourced systems (music, video, gig economy platforms), Shapley value models fair splitting of total system value between a “founder” (major participant) and “crowd” (many minor contributors). Value functions are constructed to reflect network externalities (e.g., Metcalfe’s law), with closed-form Shapley allocations derived for several models:

• Single-founder, identical crowd: With ν(S)=0 if the founder is absent and ν(S)=ρ·|S{g}|^k otherwise, each minor participant receives

$\phi_{u_i} = \frac{\nu(N) - \phi_g}{n}$

where explicit expressions are given for general power-law k, and in the large-n limit, the crowd’s total Shapley share converges to k/(k+1) of the total, typically between 1/2 and 2/3 for k∈1,2.

• Empirical validation: Analysis of real-world remuneration in platforms like Spotify, Pandora, and YouTube shows Shapley-predicted splits (55–70%) align closely with actual crowd earnings (K et al., 2023).

Supply Chains and Graph-Based Systems

In logistics and transport, the allocation of route cost in the Traveling Salesperson Game (TSG) via Shapley value is computationally hard, but well-established proxies and sampling-based estimation strategies are used in practice. The edge-based Shapley value extends the framework to allocate value directly to interactions in networks, crucial in supply chains and digital platforms, and maintains the core fairness axioms of the classical player-based Shapley value (Yamada et al., 16 Jul 2025).

3. Extensions, Variants, and Generalizations

Heterogeneity and Weighted Contributions

Standard Shapley allocation assumes agent symmetry. Heterogeneous contribution or "capability" can be incorporated by adjusting marginal contributions with exogenous weights. In the We Media value chain, an AHP-derived innovation impact factor G_i modifies each term:

$$\phi'_i(v) = \sum_{S \subseteq N \setminus \{i\}} w(S) \Big\{ [v(S \cup \{i\}) - v(S)] + v(S) [G_i - 1/n] \Big\}$$

where w(S) is the standard combinatorial weight, and G_i reflects each participant's evaluated impact (∑G_i=1) (Xu et al., 24 Dec 2024).

Incomplete and Structured Games

When coalition values are only partially specified (e.g., intersection-closed set systems), the uniform-dividend value (UD-value) provides a Shapley-type allocation by uniformly distributing undetermined surpluses over equivalence classes of coalitions. The UD-value is shown to be the expectation of the Shapley value over all positive extensions of the partial game, and its axiomatic foundation generalizes efficiency, additivity, and symmetry (Černý, 9 Jan 2025).

4. Computational Methods and Scalability

Exact Algorithms

• For games exhibiting exploitable structure (e.g., independent utilities in data marketplaces, or variance games in finance where the characteristic function decomposes into pairwise covariances), custom analytical formulas enable O(n^2) or better computation. For variance games, the Shapley value for factor i is simply:

$\phi_i(\mathrm{Var}) = \mathrm{Cov}(X_i,\,X)$

with X the portfolio sum (Colini-Baldeschi et al., 2016).

Approximation and Sampling

• Monte Carlo sampling (permutation or subset) provides unbiased estimation with error decaying as O(1/√m) in the number of samples m. Variance-reduced or regression-based techniques (e.g., KernelSHAP, SurroShap) further scale to thousands of entities by use of surrogate models (deep learning) combined with sampling and closed-form solution in the sample space, with theoretical error bounds on allocation accuracy (Feng et al., 3 Nov 2025).
• Clustering approaches (e.g., customer grouping in grid allocation) reduce the dimension of coalitions to compute, attaining high accuracy (RMSE <10^-2 for n up to 125) (Azuatalam et al., 2019).

Lower/Upper Bounds and Structural Preprocessing

• In large allocation problems, modularity, separability, and pruning based on neighborhood profiling (e.g., in research evaluation or goods assignment) allow the vast majority of Shapley values to be resolved exactly or firmly bounded without full enumeration, enabling tractable approximation in real instances with thousands of agents (Lupia et al., 2017).

5. Specialized Settings and Algorithmic Innovations

Sequential and Ordered Attribution

The ordered Shapley value method provides a refined allocation in path-dependent processes (e.g., multi-channel online advertising), attributing revenue or success at each sequential touchpoint to the appropriate channel and stage. Closed formulæ enable decomposition of marginal revenue by the occurrence/order of channel activation, facilitating nuanced attribution and campaign optimization (Zhao et al., 2018).

Negative and Absolute Contributions

When marginal contributions can be negative (e.g., harmful data or adversarial coalitions), the absolute Shapley value and zeroed variants are considered. While original (signed) Shapley value uniquely preserves group rationality, fairness, and additivity, the absolute-value variant provides the strongest discrimination of influential positive and negative contributors at the cost of budget imbalance (Liu, 2020).

Network Topology and Path Integrals

The Hodge-Shapley value extends classical allocation from grand coalition splits to all subcoalitions by interpreting allocation as a stochastic path integral or solving a Poisson equation on the coalition graph, with unique allocation at every coalition state satisfying five axioms extending Shapley's four (Lim, 2022).

6. Empirical Results and Practical Recommendations

Empirical evaluation across domains consistently shows that Shapley value allocation tracks cost/benefit causality more closely than naive or heuristic shares, respects fairness even under heterogeneous contribution regimes, and incentivizes investment and participation where it most advances system value (K et al., 2023, Park et al., 25 Oct 2025, Xu et al., 24 Dec 2024). For computational practice:

• Apply structure-exploiting reductions and cluster-based proxies wherever possible to contain exponential scaling.
• Use model surrogates or sampling-based regression for large games with expensive characteristic functions, with bias correction for fairness.
• Maintain awareness of variant choice: standard Shapley for strict budget-balance, absolute/zero-value versions where interpretability or influence magnitude is paramount.

• "Fair Allocation in Crowd-Sourced Systems" (K et al., 2023)
• "A Turvey-Shapley Value Method for Distribution Network Cost Allocation" (Azuatalam et al., 2019)
• "Variance Allocation and Shapley Value" (Colini-Baldeschi et al., 2016)
• "Absolute Shapley Value" (Liu, 2020)
• "Profit Allocation in the We Media Value Chain: A Shapley Value-Based Approach" (Xu et al., 24 Dec 2024)
• "Fair Cost Allocation in Energy Communities: A DLMP-based Bilevel Optimization with a Shapley Value Approach" (Park et al., 25 Oct 2025)
• "The Shapley value and the strength of weak players in Big Boss games" (Guardiola et al., 24 Mar 2025)
• "A New Value for Cooperative Games on Intersection-Closed Systems" (Černý, 9 Jan 2025)
• "Shapley-Value-Based Graph Sparsification for GNN Inference" (Akkas et al., 28 Jul 2025)
• "Fair Representation and a Linear Shapley Rule" (Kurz et al., 2016)
• "A Study of Proxies for Shapley Allocations of Transport Costs" (Aziz et al., 2014)
• "Deep Learning-Accelerated Shapley Value for Fair Allocation in Power Systems: The Case of Carbon Emission Responsibility" (Feng et al., 3 Nov 2025)
• "New allocation rule based on graph structures and their application to economic phenomena" (Yamada et al., 16 Jul 2025)
• "Portfolio risk allocation through Shapley value" (Hagan et al., 2021)
• "The Shapley Value in the Knaster Gain Game" (Briata et al., 2016)
• "Network analysis using Forman curvature and Shapley values on hypergraphs" (Yamada, 2021)
• "On Shapley Value in Data Assemblage Under Independent Utility" (Luo et al., 2022)
• "Computing the Shapley Value in Allocation Problems: Approximations and Bounds, with an Application to the Italian VQR Research Assessment Program" (Lupia et al., 2017)
• "Shapley Value Methods for Attribution Modeling in Online Advertising" (Zhao et al., 2018)
• "Cooperative networks and Hodge-Shapley value" (Lim, 2022)