This paper deals with Harsanyi power solutions for cooperative games in which partial cooperation is based on specific union stable systems given by the winning coalitions derived from a voting game. This framework allows for analyzing new and real situations in which there exists a feedback between the economic influence of each coalition of agents and its political power. We provide an axiomatic characterization of the Harsanyi power solutions on the subclass of union stable systems arisen from the winning coalitions from a voting game when the influence is determined by a power index. In particular, we establish comparable axiomatizations, in this context, when considering the Shapley-Shubik power index, the Banzhaf index and the Equal division power index which reduces to the Myerson value on union stable systems. Finally, a new characterization for the Harsanyi power solutions on the whole class of union stable systems is provided and, as a consequence, a characterization of the Myerson value is obtained when the equal power measure is considered.