Method of Equal Shares: Difference between revisions
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{{wikipedia}}
The '''Method of Equal Shares''' <ref name="rulex">In early papers the method has been also referred to as Rule X.</ref> (sometimes referred to as '''MES''') is a proportional method of counting ballots that applies to [[participatory budgeting]] and to [[Multi-member system|committee elections]].<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> MES ideally uses [[Rated ballot|rated ballots]]
== Motivation ==
The method is an alternative to [[knapsack voting]] which is used by most cities even though it is a disproportional method. For example, if 51% of the population support 10 red projects and 49% support 10 blue projects, and the money suffices only for 10 projects, the knapsack budgeting will choose the 10 red supported by the 51%, and completely ignore the wishes of the 49%
The method guarantees [[proportional representation]]: it satisfies the strongest known variant of the [[justified representation]] axiom that is known to be satisfiable in participatory budgeting.
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The method can be applied in two ways to the setting where the voters vote by marking the projects they like (see [[#Example 1|Example 1]]):
<ol>
<li> Setting <math> u_i(p) = \mathrm{cost}(p) </math> if project <math> p </math> is approved by voter <math> i </math>, and <math> u_i(p) = 0 </math> otherwise. This assumes that the utility of a voter equals the total amount of money spent on the projects supported by the voter. This assumption is commonly used in other methods of counting approval ballots for participatory budgeting, for example
<li> Setting <math> u_i(p) = 1 </math> if project <math> p </math> is approved by voter <math> i </math>, and <math> u_i(p) = 0 </math> otherwise. This assumes that the utility of a voter equals the number of approved selected projects. This typically results in selecting more but less expensive projects. </li>
</ol>
==== Ranked ballots ====
For ranked ballots, MES uses a method related to the [[Expanding Approvals Rule|expanding approvals rule]]. Like EAR, it passes [[Proportionality for Solid Coalitions|proportionality for solid coalitions]].
The method applies to the model where the voters vote by ranking the projects from the most to the least preferred one. Assuming [[lexicographic preferences]], one can use the convention that <math> u_i(p) </math> depends on the position of project <math> p </math> in the voter's <math> i </math> ranking, and that <math>u_i(p)/u_i(p') \to \infty </math>, whenever <math> i </math> ranks <math> p </math> as more preferred than <math> p' </math>.
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