Droop quota

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In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff quota^[a]) is the minimum number of votes needed for a party or candidate to guarantee themselves one extra seat in a legislature in modern STV and other voting systems. It is the preferred quota, being known to be less likely than the Hare quota, to give majority of seats to a minority party.^[1] It is the smallest portion of votes that elects the correct number of members to fill the seats, but no more than that number.

It generalizes the concept of a majority to multiple-winner elections: just as a majority (more than half of votes) guarantees a candidate can be declared the winner of a one-on-one election, having more than one Droop quota's worth of votes measures the number of votes a candidate needs to be guaranteed victory in a multiwinner election.

Besides establishing winners, the Droop quota is used to define the number of excess votes (votes not needed for a candidate that is declared elected). In proportional systems such as STV, CPO-STV, and proportional approval (or score) voting, these excess votes are transferred to other candidates, if possible, preventing them from being wasted.

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 ${\displaystyle k}$-winner election is any amount larger than the formula:^[b]

For simplicity and regarding voting sytems that use whole votes (not fractional votes), Droop is one more than the total number of valid votes divided by one more than the number of seats. H.R. Droop put it this way: "mV n+1 + i votes will be sufficient to elect one representative."^[2]

In the case of a single-winner election, this reduces to the familiar simple majority rule. Under such a rule, a candidate can be declared elected as soon as they have strictly more than 50% of the vote, i.e one more than. ${\textstyle {\frac {\text{total votes}}{2}}}$.

Sometimes, the Droop quota is written as a share (i.e. percentage) of the total votes.

Sometimes Droop is written as one more than 1⁄k+1, in which case the winner is a candidate who has a number of votes equal to or greater than the quota.

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 1⁄k+1 votes or fewer. 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 ${\textstyle {\frac {100}{3+1}}+1=26}$. 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 Y
• Hamilton: 10
• Burr: 20
• Jefferson: 25

Only Washington has strictly more than 25 votes. As a result, he is immediately elected. Washington has 19 excess votes that can be transferred to their second choice, Hamilton. The tallies therefore become:

• Washington: 26 Y
• Hamilton: 29 Y
• Burr: 20
• Jefferson: 25

Hamilton is elected, so his excess votes are redistributed. Thanks to Hamilton's support, Jefferson wins 28 votes to Burr's 20 and is elected.

It can happen that two candidate might be tied although with thousands of votes used in real STV elections this is extremely rare.. The tiebreaking rules discussed below indicate that the candidate with more first-preference votes is declared the winner.

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.

(Part of the difference is produced by some applications saying a winner must take more than the quota, while others say the winner must take at least the quota. Droop himself presented the Droop quota as votes/(number of seats plus 1), plus 1.)^[3]

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.

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.^[4] 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

Whenever two candidates are tied in an STV election, some systems say that 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.)

Other systems say in case of a tie, the candidate with the most first preference votes should be declared the winner.

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 right-hand limit of seat apportionments as the quota approaches the exact Droop quota from above, an approach that allows for calculating additional tiebreakers when needed (in favor of the least well-represented voters).^[c]

Confusion with the Hare quota

The Droop quota is sometimes confused with the more intuitive Hare quota.

Some of the confusion between the two quotas may result from a fencepost error, caused by forgetting that sometimes votes are left with unelected candidates at the end of the counting process. As well, under STV, it is possible for candidates to be elected with less than quota at the end of the count when the field of candidates is thinned to the number of remaining open seats. It is sometimes said that those sub-quota winners are more common under systems that use the Hare quota but Droop system can also produce them.^[5] In vote counts that fill seats with sub-quota winners, there are no candidates who are neither elected nor eliminated. Waste of votes by un-used votes is decreased but at expense of equality of winners' vote tallies. But systems that elect multiple winners based on quota, such as Droop, are more fair and more proportional than systems where single winners are elected in isolated, separate contests.

The Droop quota is today the most popular quota for STV elections.^{[citation needed]}

See also

• List of democracy and elections-related topics

Notes

References

Further reading

• Droop, Henry Richmond (1869). On the Political and Social Effects of Different Methods of Electing Representatives. London.{{cite book}}: CS1 maint: location missing publisher (link)
• Droop, Henry Richmond (1881). "On methods of electing representatives" (PDF). Journal of the Statistical Society of London. 44 (2): 141–196 [Discussion, 197–202]. doi:10.2307/2339223. JSTOR 2339223. Reprinted in Voting matters Issue 24 (October 2007) pp. 7–46.