On the Extreme Points of Two Polytopes associated with a Digraph and Applications to Cooperative Games

In this paper we describe the extreme points of two closely related polytopes that are assigned to a digraph. The first polytope is the set of all sharing vectors (elements from the unit simplex) such that each node gets at least as much as each of its successors. The second one is the set of all fuzzy vectors (elements of the unit cube) with participation rates of players subordinated to the relationships prescribed by the digraph. We also discuss some applications in cooperative game theory.