Characterization of the Shapley-Shubik Power Index Without the E ¢ ciency Axiom

We show that the Shapley-Shubik power index on the domain of simple (voting) games can be uniquely characterized without the e¢ ciency axiom. In our axiomatization, the e¢ ciency 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… (More)