Expanding Approvals Rule: Difference between revisions
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By analogy to the [[single transferable vote]], the Expanding Approvals Rule is also known as '''Bucklin transferable vote''' ('''BTV''').<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2011-October/093877.html|title=Declaration; are you ready to sign?|website=Election-methods mailing list archives|date=2011-10-01}}</ref>
There is a great deal of flexibility in how EAR is implemented (e.g., the specific quota applied, how voter weights are spent, how exhausted ballots are interpreted, and how candidates are elected when there are "ties"). Therefore, EAR may be considered as a family of voting rules rather than one specific voting.
== Semi-solid coalitions ==▼
The authors<ref name="Aziz Lee pp. 1–45" /> propose a specific EAR voting rule. Suppose $m$ is the number of candidates, $k$ is the number of candidates to be elected, and $n$ is the number of voters. The authors propose that
# EAR be applied with the quota displayed in Figure 1, which is essentially the Droop quota;
# <nowiki>Voter weights are "spent" using a uniform and fractional reweighting scheme (i.e., whenever a candidate is elected, every voter with a ballot supporting this candidate at this stage will have their current weight reduced by $\frac{W-\bar{q}}{W}$, where $W$ is the total weight of voters casting a ballot for the elected candidate); </nowiki>
# If a voter's ballot is exhausted at any stage, then, in all future stages, the voter is assumed to support all candidates;
# When multiple candidates can be elected at a given stage, then the candidate with the highest rank-maximal order is elected (where the rank-maximal order is calculated with respect to the voters' original ballots and, hence, this ordering remains fixed throughout the EAR process). Rank-maximal ordering is constructed as follows: Given 2 candidates $a$ and $b$, if candidate $a$ is the first preference of (strictly) more voters than candidate $b$, then candidate $a$ is strictly higher in the rank-maximal ordering. If candidate $a$ and $b$ have equal numbers of voter holding them in first preference, then we consider the number of voters that place $a$ and $b$ as their second preference and so on.
[[File:Screenshot 2022-01-31 at 10.40.01.png|thumb|Figure 1: The proposed quota]]
Unlike many other voting rules, EAR can be applied to ballots that include "indifferences" between candidates, i.e., voters are not forced to rank candidates as they can incorporate "ties" into their ballots.
== Properties ==
The authors'<ref name="Aziz Lee pp. 1–45" /> proposed EAR voting rule satisfies a number of properties:
* Generalized Proportionality for Solid Coalitions (PSC)
* Generalized weak-PSC
* Proportional Justified Representation
* PSC
* Weak-PSC
* Candidate monotonicity when voters have dichotomous (or approval) ballots
* Candidate monotonicity if only one candidate is to be elected
* Rank respecting candidate monotonicity
* Non-crossing candidate monotonicity.
Semi-solid coalitions are groups of voters who rank some candidates above all others, but some voters in the group may rank other candidates above the candidates the group ranked above all others. EAR is claimed by its creators to always elect from Droop semi-solid coalitions.<ref name="Aziz Lee pp. 1–45">{{cite journal | last=Aziz | first=Haris | last2=Lee | first2=Barton E. | title=The expanding approvals rule: improving proportional representation and monotonicity | journal=Social Choice and Welfare | publisher=Springer Science and Business Media LLC | volume=54 | issue=1 | date=2019-08-09 | issn=0176-1714 | doi=10.1007/s00355-019-01208-3 | pages=1–45 | url=https://arxiv.org/abs/1708.07580}}</ref>
According to the paper, <blockquote>(4) EAR addresses a criticism of Tideman (2006):<ref name="Tideman 2006 p.
Example 7 Consider the profile with 9 voters and where k [number of winners] = 3.
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In the example, {e1, e2, e3} is the outcome of QBS. Although PSC is not violated for voters in {1, 2, 3} but the outcome appears to be unfair to them because they almost have a solid coalition. Since they form one-third of the electorate they may feel that they deserve that at least one candidate such as c1, c2 or c3 should be selected. In contrast, it was shown in Example 5 that EAR does not have this flaw and instead produces the outcome {e1, e2, c1}.</blockquote>
==
EAR fails the monotonicity criterion.<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2018-February/001682.html|title=Path dependence monotonicity failure in BTV|website=Election-methods mailing list archives|date=2018-02-18|last=Munsterhjelm|first=K.}}</ref>
The example has two factions support two candidates each (e.g. one faction supports candidates X and Y, and another candidates W and Z), where X and W are almost exactly tied. In the original two-seat election, X wins and then the other winner must come from the WZ coalition, so Z wins. Then Z is raised on a X>Z ballot, which makes W win the first seat instead. Now the second winner must come from the other coalition, and so Y wins: raising Z makes Z lose.
==Notes==
A 2-winner example is included in Figure 2.
[[File:Screen Shot 2021-05-24 at 8.27.33 AM.png|thumb|Figure 2: A 2 winner example using the authors' proposed EAR method.]]
==See also==
*[[Evaluative Proportional Representation]]
*[[Maximum Constrained Approval Bucklin]]
▲== References ==
[[Category:Multi-winner voting methods]]
[[Category:PSC-compliant voting methods]]
[[Category:Ranked voting methods]]
<references />
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