In this paper, we consider the relationship between the Weber set and the Shapley set being the set of all weighted Shapley values of a TU-game. In particular, we propose a new proof for the fact that the Weber set always includes the Shapley set. It is shown that the inclusion mentioned follows directly from the representation theorem for the Weber set, established by Vasil'ev and van der Laan (2002), Siberian Adv. Math., V.12, N2, 97â125. Since the representation theorem applied is formulated in terms of the dividend sharing systems belonging to the so-called Weber polyhedron, we pay strong attention to some monotonicity properties of this polyhedron. Specifically, by making use of induction techniques, a new proof of the strong monotonicity of the Weber d-systems is obtained, and a simplified description of the Weber polyhedron is given.
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