# Hodge decomposition and the Shapley value of a cooperative game

arXiv: https://arxiv.org/abs/1709.08318 · HTML: https://mlsanity.com/paper/1709.08318/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2017.

## Built on by (most cited first, 2 of 2)

- [Feature Synergy, Redundancy, and Independence in Global Model Explanations using SHAP Vector Decomposition](https://mlsanity.com/paper/2107.12436/) (2021)
- [Shapley Sets: Feature Attribution via Recursive Function Decomposition](https://mlsanity.com/paper/2307.01777/) (2023)

## Abstract (arXiv)

We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.