Hodge decomposition and the Shapley value of a cooperative game

TL;DR

This paper introduces a novel decomposition of cooperative games using Hodge theory, providing a new characterization of the Shapley value and extending the framework to weighted and constrained games.

Contribution

It applies combinatorial Hodge decomposition to cooperative games, offering a new perspective on the Shapley value and generalizing to weighted and constrained coalition scenarios.

Findings

Decomposition satisfies efficiency, null-player, symmetry, and linearity properties.

Shapley value characterized as the value of the grand coalition in each component game.

Extension to weighted games and games with coalition constraints.

Abstract

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.