Droop quota
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Besides establishing winners, the Droop quota is used to define the number of
The Droop quota was first devised by the English lawyer and mathematician Henry Richmond Droop (1831–1884), as an improvement to the earliest proposals for the single transferable vote (using the Hare quota). It was later independently used by Swiss physicist Eduard Hagenbach-Bischoff for efficient calculation by the D'Hondt method.
Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Northern Ireland, Malta, and Australia. It is also used in South Africa to allocate seats by the largest remainder method.
Standard Formula
The exact form of the Droop quota for a -winner election is given by the formula:[b]
Sometimes, the Droop quota is written as a share (i.e. percentage) of the total votes, in which case it has value 1⁄k+1.
Derivation
The Droop quota can be derived by considering what would happen if k candidates (called "Droop winners") have exceeded the Droop quota; the goal is to identify whether an outside candidate could defeat any of these candidates.
In this situation, each quota winner's share of the vote exceeds 1⁄k+1, while all unelected candidates' share of the vote, taken together, is less than 1⁄k+1 votes. Thus, even if there were only one unelected candidate who held all the remaining votes, they would not be able to defeat any of the Droop winners.
Example in STV
The following election has 3 seats to be filled by single transferable vote. There are 4 candidates: George Washington, Alexander Hamilton, Thomas Jefferson, and Aaron Burr. There are 102 voters, but two of the votes are spoiled.
The total number of valid votes is 100, and there are 3 seats. The Droop quota is therefore . These votes are as follows:
45 voters | 20 voters | 25 voters | 10 voters | |
---|---|---|---|---|
1 | Washington | Burr | Jefferson | Hamilton |
2 | Hamilton | Jefferson | Burr | Washington |
3 | Jefferson | Washington | Washington | Jefferson |
First preferences for each candidate are tallied:
- Washington: 45
- Hamilton: 10
- Burr: 20
- Jefferson: 25
Only Washington has strictly more than 25 votes. As a result, he is immediately elected. Washington has 20
- Washington: 25
- Hamilton: 30
- Burr: 20
- Jefferson: 25
Hamilton is elected, so his excess votes are redistributed. Thanks to Hamilton's support, Jefferson wins 30 votes to Burr's 20 and is elected.
If Hamilton's supporters had instead backed Burr, the election for the last seat would have been exactly tied, instead of a clear win for Jefferson. The tiebreaking rules discussed below would choose Jefferson, as he earned a full Droop quota of first-preference votes.
Incorrect versions
Off-by-one errors
There is a great deal of confusion among legislators and political observers about the exact definition of the Droop quota. At least six different mistaken versions appear in various legal codes or definitions of the quota, all varying from the above definition by at most one or two votes.
The first two variants, L1 and L2, approximate the Droop quota by rounding up (to avoid decimals), and are sometimes called the rounded Droop quota.[b] These versions are sometimes used by legislators who believe a quota of votes must be a whole number. The L3 quota is caused by mistakenly ignoring the floor function in L1.
The origins of the third variant, C1, are not clear, as this variant is not original to Droop.[1] Variant S2 is sometimes smaller than the actual Droop quota, and Variant S1 is always no larger than the correct formula. In cases where they are smaller, it would be possible for them to result in too many candidates being elected.
Spoiled ballots should not be included when calculating the Droop quota; however, some jurisdictions fail to specify this in their election administration laws.
Handling ties
Some of the nonstandard formulations shown above have been justified by claiming the exact Droop quota can elect more candidates than there are seats, or that it can result in ties. However, this is incorrect, so long as candidates are only considered to be elected when their vote total is strictly greater than the Droop quota. In addition, tied votes can occur with any quota.
Whenever two candidates are tied in an STV election, ties should be broken by ignoring ballots transferred from previous winners. In other words, candidates should be ordered first by their total number of votes, and then by the number of votes they have that have never used to elect a winner. (This should not be confused with ordering candidates by their number of first-preference votes, as votes transferred after a candidate has been eliminated should still be included in the vote total.)
This rule has the advantage of minimizing the number of voters with no representation (i.e. whose ballots are not used to elect any candidate). It can also be justified by taking the
Confusion with the Hare quota
The Droop quota is often confused with the more intuitive Hare quota. Unlike the Droop quota, which gives the votes needed to elect a candidate, the Hare quota gives the number of voters who are represented by a single candidate.
The confusion between the two quotas originates results from a
The Droop quota is today the most popular quota for STV elections.[citation needed]
See also
- List of democracy and elections-related topics
Notes
- ^ Some texts distinguish between a "Droop quota" and a "Hagenbach-Bischoff quota." (Here, the Droop quota is defined as the Hagenbach-Bischoff quota rounded to the next-largest integer).
- ^ a b Some authors use the terms "Hagenbach-Bischoff quota" or "exact Droop quota" to refer to the quantity described in this article, and reserve the term "Droop quota" for the rounded Droop quota (the original form in the works of Henry Droop).
- ^ This procedure gives a "leximax" ordering, ranking candidates by the number of ballots previously used to elect only (0, 1, 2...) candidates.
References
- ^ Dančišin, Vladimír (2013). "Misinterpretation of the Hagenbach-Bischoff quota". Annales Scientia Politica. 2 (1).
Further reading
- Droop, Henry Richmond (1869). On the Political and Social Effects of Different Methods of Electing Representatives. London.
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: CS1 maint: location missing publisher (link) - Droop, Henry Richmond (1881). "On methods of electing representatives" (PDF). JSTOR 2339223. Reprinted in Voting mattersIssue 24 (October 2007) pp. 7–46.