Method of Equal Shares: Difference between revisions
Updated some of the links and added a bit about SSS as a possible completion
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'''The Method of Equal Shares'''<ref name="rulex">In early papers the method has been also referred to as Rule X.</ref> is a proportional method of counting ballots that applies to [[participatory budgeting]] and to [[
<ref>{{Cite journal|last1=Peters|first1=Dominik|last2=Skowron|first2=Piotr|title=Proportionality and the Limits of Welfarism|journal=Proceedings of the 21st ACM Conference on Economics and Computation|series=EC'20|year=2020|pages=793–794|doi=10.1145/3391403.3399465|arxiv=1911.11747|isbn=9781450379755|url=https://arxiv.org/abs/1911.11747}}</ref>
<ref>{{Cite journal|last1=Pierczyński|first1=Grzegorz|last2=Peters|first2=Dominik|last3=Skowron|first3=Piotr|title=Proportional Participatory Budgeting with Additive Utilities.|journal=Proceedings of the 2021 Conference on Neural Information Processing Systems|series=NeurIPS'21|year=2020|arxiv=2008.13276|url=https://arxiv.org/abs/2008.13276}}</ref> It can be used, when the voters vote via [[approval ballot|approval ballots]], [[Ranked ballot|ranked ballots]] or [[cardinal voting|cardinal ballots]].
== Motivation ==
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== Definition ==
This section presents the definition of the rule for [[cardinal voting|cardinal ballots]]. See [[#Discussion|discussion]] for a discussion on how to apply this definition to [[approval ballot|approval ballots]] and [[Ranked ballot|ranked ballots]].
We have a set of projects <math>P = \{p_1, p_2, \ldots, p_m\}</math>, and a set of voters <math>N = \{1, 2, \ldots, n\}</math>. For each project <math>p \in P</math> let <math>\mathrm{cost}(p)</math> denote its cost, and let <math>b</math> denote the size of the available municipal budget. For each voter <math>i \in N</math> and each project <math>p \in P</math> let <math>u_i(p) </math> denote the <math>i</math>'s cardinal ballot on <math>p</math>, that is the number that quantifies the level of appreciation of voter <math>i</math> towards project <math>p</math>.
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=== Committee elections ===
In the context of [[
=== Unspent budget ===
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<ol>
<li> The utilitarian method ([[Bloc voting|Bloc Score Voting]]): the projects <math>p</math> are selected in the order of <math> \frac{\sum_{i \in N}u_i(p)}{\mathrm{cost}(p)} </math> until no further project can be selected within the budget limit. </li>
<li> Adjusting initial budget: the initial budget can be adjusted to the highest possible value which makes the method select projects, whose total cost does not exceed the unadjusted budget. </li>
<li> Other Cardinal PR systems: The remaining winners can be found with the use of other [[Cardinal PR]] systems. [[Sequentially Spent Score]] would be a natural completion since it maintains the concept of spending. </li>
</ol>
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