Journal Article

Decomposition of solutions and the Shapley value

Games and Economic Behavior
André Casajus, Frank Huettner
Subject(s): Management sciences, decision sciences and quantitative methods
Keyword(s): Decomposition, Shapley value, Potential, Consistency, Higher-order contributions, Balanced contributions
JEL Code(s): C71, D60

We suggest foundations for the Shapley value and for the naïve solution, which assigns to any player the difference between the worth of the grand coalition and its worth after this player left the game. To this end, we introduce the decomposition of solutions for cooperative games with transferable utility. A decomposer of a solution is another solution that splits the former into a direct part and an indirect part. While the direct part (the decomposer) measures a player's contribution in a game as such, the indirect part indicates how she affects the other players' direct contributions by leaving the game. The Shapley value turns out to be unique decomposable decomposer of the naïve solution.