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The Banzhaf and Shapley-Shubik power indices were first introduced to measurethe power of voters in a weighted voting system. Given a weighted votingsystem, the fixed point of such a system is found by continually reassigningeach voter's weight with its power index until the system can no longer bechanged by the operation. We characterize all fixed points under theShapley-Shubik power index of the form $(a,b,\ldots,b)$ and give an algebraicequation which can verify in principle whether a point of this form is fixedfor Banzhaf; we also generate Shapley-Shubik fixed classes of the form$(a,a,b,\ldots,b)$. We also investigate the indices of divisor voting systemsof abundant numbers and prove that the Banzhaf and Shapley-Shubik indicesdiffer for some cases.

On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems