In this paper it is shown that the core and the bargaining sets of Davis-Maschler and Zhou coincide in a class of shortest path games.

A set of necessary and sufficient conditions for convexity of a transferable utility game in terms of its decomposition into unanimity games is shown to be minimal: none of the conditions is redundant. The result is used to provide an axiomatization of the Shapley value on the set of convex games.