The extant literature on positional power in hierarchies is mainly restricted to the analysis of power in terms of the authority structure. This is based upon the bare positions of actors and ignores the existence of decision-making mechanisms. The few studies, which also take into account decision-making mechanisms, are 'membership-based'. In an earlier paper we demonstrated that such an approach is, in general, inappropriate. As a solution we suggested an 'action-based' approach and developed a power score and measure ascribing positional power to actors in hierarchies. In this paper we axiomatize this power score and measure and illustrate their applicability. © 2011 The Author(s).