Characterization of the Shapley–Shubik power index without the efficiency axiom

We show that the Shapley–Shubik power index on the domain of simple (voting) games can be uniquely characterized without the efficiency axiom. In our axiomatization, the efficiency is replaced by the following weaker requirement that we term the gain-loss axiom: any gain in power by a player implies a loss for someone else (the axiom does not specify the extent of the loss). The rest of our axioms are standard: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, and dummy.