Power on digraphs

It is assumed that relations between $n$ players are represented by a directed graph or digraph. Such a digraph is called invariant if there is a link (arc) between any two players between whom there is also a directed path. We characterize a class of power indices for invariant digraphs based on four axioms: Null player, Constant sum, Anonymity, and the Transfer property. This class is determined by $2n-2$ parameters. By considering additional conditions about the effect of adding a directed link between two players, we single out three different, one-parameter families of power indices, reflecting several well-known indices from the literature: the Copeland score, $\beta$- and apex type indices.