A method for estimating the core of root game

It is shown that the base of the grand (shadow) subcore coincides with the core of the root game in any TU-cooperative game. Comparing the definitions of the grand subcore and the grand shadow subcore with the description of the aggregate-monotonic core leads to formal geometrical coincidence of the aggregate-monotonic core with either the grand subcore or the grand shadow subcore. The method for estimating the simplest set of equations and inequalities describing the core of a root game in a TU-game with any number of players ($n>3$) is proposed. To develop the method the duality theory and an inductive method by B. Peleg are used.