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 an additional seat in a legislature. 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 by a candidate who has been declared elected. In proportional quota-rule systems such as STV and CPO-STV, these excess votes are transferred to other candidates, preventing them from being wasted.

The Droop quota was first devised by the English lawyer and mathematician Henry Richmond Droop (1831–1884), as a replacement for the Hare quota.

Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Northern Ireland, Malta, and Australia.^{[citation needed]} It is also used in South Africa to allocate seats by the largest remainder method.^{[citation needed]}

Standard Formula

The exact form of the Droop quota for a ${\displaystyle k}$-winner election is given by the formula:^[b]

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. ${\textstyle {\frac {\text{total votes}}{2}}}$.

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.

Any candidate who attains quota or exceeds it is declared elected.

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 ${\textstyle {\frac {100}{3+1}}=25}$, plus 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: 25 Y
• Hamilton: 29Y
• Burr: 20
• Jefferson: 25

Hamilton is elected, so his excess votes are redistributed. Thanks to Hamilton's support, Jefferson receives 29 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 has more first-preference votes than Burr.

Incorrect or nonstandard variants

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 variaton seems to arise from whether the quota must be exceeded or only equalled.

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 forgetting 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 exact Droop quota (resulting in more candidates being elected than there are seats), while Variant S1 is always smaller 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 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 often confused with the more intuitive Hare quota. The Droop quota which gives the number of voters who, taken together, can force a candidate's election, while the Hare quota gives the number of voters who are represented by each candidate. If Hare quota is used, any voting block that has that number of votes will elect the member and be represented by him or her.

The confusion between the two quotas originates results from a fencepost error, caused by forgetting that unelected candidates can also have votes at the end of the counting process. In the case of a single-winner election, using the Hare quota would lead to the conclusion that a candidate must receive 100% of the vote to be certain of victory; in reality, any votes exceeding a bare majority are excess votes.

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.