Method of Equal Shares: Difference between revisions

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{{wikipedia}}
{{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 uses [[Rated ballot|rated ballots]]; the [[Expanding Approvals Rule]] allows for ranked ballots to be used instead, though expanding approvals works better when using cardinal ballots.
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]], but the authors also detail ways to use MES with ranked or approval ballots.


== Motivation ==
== 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 ignore the 49% altogether.<ref name=":4">{{Cite journal|last1=Fluschnik|first1=Till|last2=Skowron|first2=Piotr|last3=Triphaus|first3=Mervin|last4=Wilker|first4=Kai|date=2019-07-17|title=Fair Knapsack|journal=Proceedings of the AAAI Conference on Artificial Intelligence|language=en|volume=33|pages=1941–1948|doi=10.1609/aaai.v33i01.33011941|issn=2374-3468|doi-access=free}}</ref> In contrast, the method of equal shares would pick 5 blue and 5 red projects.
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%.<ref name=":4">{{Cite journal|last1=Fluschnik|first1=Till|last2=Skowron|first2=Piotr|last3=Triphaus|first3=Mervin|last4=Wilker|first4=Kai|date=2019-07-17|title=Fair Knapsack|journal=Proceedings of the AAAI Conference on Artificial Intelligence|language=en|volume=33|pages=1941–1948|doi=10.1609/aaai.v33i01.33011941|issn=2374-3468|doi-access=free}}</ref> In contrast, the method of equal shares would pick 5 blue and 5 red projects.


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.
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]]):
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>
<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 in the [[Participatory_budgeting_algorithm#Knapsack_budgeting|knapsack algorithm]], and typically results in selecting fewer more expensive projects. </li>
<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 with [[knapsack voting]], and typically results in selecting fewer more expensive projects. </li>
<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>
<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>
</ol>


==== Ranked ballots ====
==== 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>.
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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