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arXiv · 1709.08318

Hodge decomposition and the Shapley value of a cooperative game

Ari Stern·Alexander Tettenhorst
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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.

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Ari Stern, Alexander Tettenhorst. 2017-09-25. Hodge decomposition and the Shapley value of a cooperative game. https://doi.org/10.1016/j.geb.2018.09.006

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