Droop quota

In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff quota) is the minimum number of votes needed for a party or candidate to guarantee themselves one extra 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 for a candidate to be ensured victory). In proportional systems such as STV, CPO-STV, and proportional approval (or score) voting, 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 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 in the calculation of divisors for 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 $k$ -winner election is given by the formula:^[a] ${\frac {\text{total votes}}{k+1}}$ 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 $.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄k+1$ .

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 ${\textstyle {\frac {100}{3+1}}=25}$ . 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: 15
Jefferson: 25

Only Washington has strictly more than 25 votes. As a result, he is immediately elected. Washington has 20 excess votes that can be transferred to their second choice, Hamilton. The tallies therefore become:

Washington: 25 Y
Hamilton: 30 Y
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.^[a] 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 $S2$ can sometimes be smaller than the actual Droop quota, and Variant $S1$ is always no larger than the correct version's formula, however.The final two variants are smaller than the correct quota, and can result in too many candidates being elected. ${\begin{array}{rlrlrl}{\text{L1:}}&&{\Bigl \lfloor }{\frac {\text{total votes}}{{\text{seats}}+1}}{\Bigr \rfloor }+1&&{\text{L2:}}&&\left\lceil {\frac {\text{total votes}}{{\text{seats}}+1}}\right\rceil &&{\text{L3:}}&&{\frac {\text{total votes}}{{\text{seats}}+1}}+1\\{\text{C1:}}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{total votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}&&{\text{S1:}}&&\left\lfloor {\frac {\text{total votes}}{{\text{seats}}+1}}\right\rfloor &&{\text{S2:}}&&\left\lfloor {\frac {\text{total votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor \end{array}}$ 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).^[b]

Confusion with the Hare quota

The Droop quota is often confused with the more intuitive Hare quota, which is used mostly to apportion seats according to the largest remainder method (where it gives unbiased results). Unlike the Droop quota, which calculates the number of voters who elect a given candidate, the Hare quota (seats-to-votes ratio) gives the number of voters who are represented by a single candidate.

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.

Under the Hare quota, a large number of votes (a number equal to the Droop quota) end up wasted; most of these come from the highest-ranked winners. Results with the Hare quota tend to be disproportionate, with a notable bias towards smaller parties, as the last-place finisher has the opportunity to elect a candidate with fewer than one Hare quota's worth of votes.

The earliest writings of Thomas Hare assumed the use of the Hare quota for STV; since then, mathematicians and political scientists have concluded that "the Droop quota is to be preferred to the Hare quota."^[2] The Droop quota is today the most popular quota for STV elections.

Notes

^ ^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).
^ Papers and Proceedings of the Royal Society of Tasmania. Royal Society of Tasmania. 1913.

v t e Types of majority and other electoral quotas
Single winner (majority)	Plurality Majority Absolute majority Supermajority Double majority
Electoral quotas	Hare quota Droop quota Imperiali quota (pseudoquota)
Other	Electoral threshold