Equivalent N-Person Games and the Null Space of the Shapley Value

Two n-person games may be said to be equivalent if they have the same Shapley value. In this paper, we attempt to simplify the determination of the equivalence of two games. We do this by recognizing that a pair of games are equivalent if and only if their difference lies in the null space of the Shapley value. Using the representation theory of the symmetric groups we construct a direct-sum decomposition of this null space into invariant subspaces. We then use this same theory to derive a characterization of a very general type of value, of which the Shapley value is one particular example.