On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems
The Banzhaf and Shapley-Shubik power indices were first introduced to measure the power of voters in a weighted voting system. Given a weighted voting system, the fixed point of such a system is found by continually reassigning each voter's weight with its power index until the system can no lo...
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| Main Authors: | , , , , , |
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| Format: | Journal Article |
| Language: | English |
| Published: |
16-10-2020
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| Subjects: | |
| Online Access: | Get full text |
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| Summary: | The Banzhaf and Shapley-Shubik power indices were first introduced to measure
the power of voters in a weighted voting system. Given a weighted voting
system, the fixed point of such a system is found by continually reassigning
each voter's weight with its power index until the system can no longer be
changed by the operation. We characterize all fixed points under the
Shapley-Shubik power index of the form $(a,b,\ldots,b)$ and give an algebraic
equation which can verify in principle whether a point of this form is fixed
for Banzhaf; we also generate Shapley-Shubik fixed classes of the form
$(a,a,b,\ldots,b)$. We also investigate the indices of divisor voting systems
of abundant numbers and prove that the Banzhaf and Shapley-Shubik indices
differ for some cases. |
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| DOI: | 10.48550/arxiv.2010.08672 |