A new basis and the Shapley value

•We introduce a new basis of the set of all TU games.•The coefficients in the linear combination of the basis coincide with the Shapley value.•The basis induces the null space of the Shapley value.•The basis is applicable to axiomatization or the inverse problem of the Shapley value.

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game in which cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.