The Orthogonal Decomposition of Games and an Averaging Formula for the Shapley Value

In this paper we attempt to decompose a game v into two different components, one lying in the null space of the Shapley value and the other in its orthogonal complement. We observe that the Shapley value of the former must be 0, so that the Shapley value of the latter coincides with the value of the original game. In this way we arrive at a new explicit formula for the Shapley value which, unlike the typical one, involves averages of player worths across coalition sizes. Central to our ideas is the game-theoretic contrast between the spaces in which each component lies.