The apex power measure for directed networks

In this paper we provide an axiomatization of the Shapley value restricted to the class of apex games using an equal loss property which states that the payoff of an apex player and a non-apex player decrease by the same amount if we make this particular non-apex player a null player. We also generalize this axiomatization to the class of games that can be obtained as sums of apex games. After discussing these axiomatizations we apply apex games and their Shapley values in measuring relational power in directed networks. We conclude by mentioning how these results can be adapted to give axiomatizations of the Banzhaf value.