Nodes of directed graphs ranked by solutions defined on cooperative games

Hierarchical structures, transportation systems, communication networks and even sports competitions can be modeled by means of directed graphs. Since digraphs without a predefined game are considered, the main part of the work is devoted to establish conditions on cooperative games so that they can be used to measure accessibility to the nodes. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given for every ordered coalition. Solutions here proposed are extensions of the wide family of semivalues to games in generalized characteristic function form.Postprint (published version