Expanding Approvals Rule

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The Expanding Approvals Rule (EAR), can be thought of as a sequential PR version of Bucklin Voting which can use the Droop quota. It works by first examining the 1st rank; any candidate with a Droop quota of 1st choices is elected, and a Droop quota of the ballots ranking them 1st are spent. Then, it examines both the 1st and 2nd ranks; any candidate ranked either 1st or 2nd by a Droop quota of ballots is elected, and a Droop quota of the ballots ranking them either 1st or 2nd are spent. And so on, until all ranks have been examined.

By analogy to the single transferable vote, the Expanding Approvals Rule is also known as Bucklin transferable vote (BTV).[1]

Semi-solid coalitions[edit | edit source]

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.[2]

According to the paper,
(4) EAR addresses a criticism of Tideman (2006):[3] “Suppose there are voters who would be members of a solid coalition except that they included an “extraneous” candidate, which is quickly eliminated among their top choices. These voters’ nearly solid support for the coalition counts for nothing which seems to me inappropriate.” We demonstrate the last flaw of QBS pointed out by Tideman in the explicit example below. EAR does not have this flaw.

Example 7 Consider the profile with 9 voters and where k [number of winners] = 3.

1 : c1, c2, c3, e1, e2, e3, e4, d1

2 : c2, c3, c1, e1, e2, e3, e4, d1

3 : c3, c1, d1, c2, e1, e2, e3, e4

4 : e1, e2, e3, e4, c1, c2, c3, d1

5 : e1, e2, e3, e4, c1, c2, c3, d1

6 : e1, e2, e3, e4, c1, c2, c3, d1

7 : e1, e2, e3, e4, c1, c2, c3, d1

8 : e1, e2, e3, e4, c1, c2, c3, d1

9 : e1, e2, e3, e4, c1, c2, c3, d1

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}.

Monotonicity[edit | edit source]

EAR fails the monotonicity criterion.[4]

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[edit | edit source]

2-winner example:

10: A>B>C

10: D>B>C

6 E

6 F

EAR elects B and C here. Yet arguably A and D are better from the perspective of PSC, since they are the 1st choices of the voters represented by B and C.

One possible way to elect A and D here might be to somehow use EAR to apportion seats to groups of voters (i.e. guarantee that the 20 B>C voters will get both seats on account of them being able to split into two groups of 10, larger than any other group), then rerun the election (with a new reduced quota, A and D are each the 1st choices of half of the voters and would thus win). Such an idea might be best implemented by using a highest averages method to reweight ballots, similar to SPAV.

Quota could be reduced by accounting for exhausted ballots. Quota = (total preferences in first n ranks)/(n*(k+1))

See also[edit | edit source]

References[edit | edit source]

  1. "Declaration; are you ready to sign?". Election-methods mailing list archives. 2011-10-01.
  2. Aziz, Haris; Lee, Barton E. (2019-08-09). "The expanding approvals rule: improving proportional representation and monotonicity". Social Choice and Welfare. Springer Science and Business Media LLC. 54 (1): 1–45. doi:10.1007/s00355-019-01208-3. ISSN 0176-1714.
  3. Tideman, N. (2006). Collective Decisions and Voting: The Potential for Public Choice. Ashgate. ISBN 978-0-7546-4717-1. Retrieved 2020-02-05.
  4. Munsterhjelm, K. (2018-02-18). "Path dependence monotonicity failure in BTV". Election-methods mailing list archives.