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.