ABSTRACT

The purpose of this chapter is to introduce new bases and explore their properties. We extend the basis consisting of the commander games introduced by Yokote, Funaki and Kamijo and provide new mathematical tools for analyzing the Shapley value. Our new game, which we call the (T,k)-intermediate game, assigns 1 to a coalition including k players in a fixed coalition T and 0, otherwise. The commander games are (T,1)-intermediate games. We show that, if some relationship between the size of coalition T and k holds, then we can construct a basis. All the new bases preserve two desirable properties of the commander games. When we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value, and the basis induces the null space of the Shapley value. We apply our basis to the analysis of coincidence conditions between the Shapley value and the prenucleolus. Our basis enables us to take a linear algebraic approach to a coincidence condition known as the PS property of Kar, Mitra and Mutuswami and clarify the mathematical structure behind the coincidence region.