# Droop quota: Wikis

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# Encyclopedia

The Droop Quota is the quota most commonly used in elections held under the Single Transferable Vote (STV) system. It is also sometimes used in elections held under the largest remainder method of party-list proportional representation (list PR). In an STV election the quota is the minimum number of votes a candidate must receive in order to be elected. Any votes a candidate receives above the quota are transferred to another candidate. The Droop quota was devised in 1868 by the English lawyer and mathematician Henry Richmond Droop (1831-1884) as a replacement for the earlier Hare quota.

Today the Droop quota is used in almost all STV elections, including the forms of STV used in the Republic of Ireland, Northern Ireland, Malta and Australia, among other places. The Droop quota is very similar to, but distinct from, the simpler Hagenbach-Bischoff quota, which is also sometimes loosely referred to as the 'Droop quota'.

## Formula

Sources differ as to the exact formula for the Droop quota. As used in the Republic of Ireland the formula is usually written:

$\left( \frac{\mbox{Total Valid Poll}}{\left(\mbox{Seats}+1 \right)} \right) + 1$
• Total Valid Poll = Total number of valid (unspoilt) votes cast in an election.
• Seats = Total number of seats to be filled in the election.

(The extra parentheses, while not strictly necessary from a mathematical standpoint, are often included in order to make the formula seem less ambiguous to non-mathematicians—if calculated out of sequence, an incorrect result would be arrived at, producing an incorrect quota.) It is important to use the Total Valid Poll, which is arrived at by subtracting the spoiled and invalid votes from the total poll.

The Droop quota is the smallest number that guarantees that no more candidates can reach the quota than the number of seats available to be filled. From a mathematical point of view, the Droop quota is the integral part of the (positive) number $\frac{\mbox{Votes}}{\mbox{Seats}+1}+1$ (see also floor and ceiling functions). This gives the Droop quota the special property that it is the smallest integral quota which guarantees that the number of candidates able to reach this quota cannot exceed the number of seats. In a single winner election, in which STV becomes the same as Instant Run-off Voting, the Droop quota becomes a simple integral majority quota–that is, it will be equal to an absolute majority of votes.

While in theory every STV election should see the right number of candidates elected through reaching the quota, in practice many voters may only vote for a small proportion of the candidates on the ballot paper, such as only those candidates from one party, or even only one candidate. Those votes are known as 'NTs', or 'non transferable votes', and the effect of their removal from the total valid poll may be to reduce the total number of votes available to such an extent that the last candidate left in a race may not actually have enough votes to reach the quota. Nevertheless, in reality, as no other candidate may mathematically be able to overtake them as the candidate nearest to the quota, they may in such circumstances be deemed elected "without reaching the quota". The quota is in fact constructed to ensure that it is mathematically impossible for candidates to achieve the quota beyond the number of available seats.

## An example

To see how the Droop quota works in an STV election imagine an election in which there are 2 seats to be filled and 3 candidates: Andrea, Carter and Brad. There are 102 voters. Two of these voters spoil their ballot papers. The remaining 100 voters vote as follows:

 45 voters Andrea Carter 25 voters Carter 30 voters Brad

There are 102 voters but two spoil their papers so the Total Valid Poll is 100. There are 2 seats. Before rounding down the Droop quota is therefore:

$\frac{100}{2+1} + 1 = 34 \frac{1}{3}$

Rounded down to the nearest integer the Droop quota is found to be 34. To begin the count the first preferences cast for each candidate are tallied and are as follows:

• Andrea: 45
• Carter: 25

Andrea has more than 34 votes. She therefore has reached the quota and is declared elected. She has 11 votes more than the quota so these votes are transferred to Carter. The tallies therefore become:

• Carter: 36

Carter now has reached the quota so is declared elected. The winners of the election are therefore Andrea and Carter.

## Advantage over the Hare quota

The earliest versions of STV used the Hare quota. The Hare quota is equal to the total valid poll divided by the total number of seats. The Droop quota is generally considered superior to the Hare quota for two reasons. First, because it elects more candidates in the first distribution of seats (whether STV or list PR) than is the case with the Hare quota, so that the maximum number of candidates are elected by the full quota without the even theoretical risk, carried by smaller quotas, of more candidates being elected than there are seats to be filled. Second, because under the Hare quota it is sometimes possible for a group of candidates supported by a majority of voters to receive only a minority of seats, and this result is considered undemocratic. This is best illustrated by an example.

### Scenario

Imagine an election in which there are 5 seats to be filled. There are 6 candidates divided between two groups: Andrea, Carter and Brad are members of the Alpha party; Delilah, Scott and Jennifer are members of the Beta party. There are 120 voters and they vote as follows:

Alpha party Beta party

31 voters

1. Andrea
2. Carter

30 voters

1. Carter
2. Andrea

2 voters

2. Andrea
3. Carter

20 voters

1. Delilah
2. Scott
3. Jennifer

20 voters

1. Scott
2. Delilah
3. Jennifer

17 voters

1. Jennifer
2. Delilah
3. Scott

It can be seen that supporters of the Alpha party all rank all three Alpha party candidates higher than any of the Beta party candidates (the final three preferences of the voters are not shown above because they will not affect the result of the election). Similarly, voters who support the Beta party all give their first three preferences to Beta party candidates. Overall, the Alpha party receives 63 votes out of a total of 120 votes. The Alpha party therefore has a majority of about 53%. The Beta party receives a 47% share of the vote.

Below the election results are shown first under the Hare quota and then under the Droop quota. It can be seen that under the Hare quota, despite receiving 53% of the vote, the Alpha party receives only a minority of seats. When the same election is conducted under the Droop quota, however, the Alpha party's majority is rewarded with a majority of seats.

### Count under the Hare quota

1. The Hare quota is calculated as 24.

2. When first preferences are tallied Andrea and Carter have both reached a quota and are declared elected. Andrea has a surplus of 7 and Carter has a surplus of 6. Both surpluses are transferred to Brad (who is of the same party) so the tallies become:

• Delilah (Beta party): 20
• Scott (Beta party): 20
• Jennifer (Beta party): 17

4. No candidate has reached a quota. Brad is the candidate with the fewest votes and so he is excluded. Because just three candidates remain and there are only three more seats to be filled, Delilah, Scott and Jennifer are all declared elected.

Result: The elected candidates are: Andrea and Carter (from the Alpha party), and Delilah, Scott and Jennifer (from the Beta party).

### Count under the Droop quota

1. The Droop quota is calculated as 21.

2. When first preferences are tallied Andrea and Carter have reached the quota and, as before, are declared elected. However this time Andrea has a surplus of 10 and Carter a surplus of 9. These surpluses transfer to Brad and the tallies become:

• Delilah (Beta party): 20
• Scott (Beta party): 20
• Jennifer (Beta party): 17

3. Brad has now reached a quota and is declared elected. He has no surplus so Jennifer, who this time has the fewest votes, is excluded. Because only Delilah and Scott are left in the count, and there are only two seats left to fill, they are both declared elected.

Result: The elected candidates are Andrea, Carter and Brad (from the Alpha party) and Delilah and Scott (from the Beta party).

## Comparison to Hagenbach-Bischoff quota

The Droop quota does not absolutely guarantee that a party with the support of a solid majority of voters will not receive a minority of seats. The only quota under which this cannot happen, even in rare cases, is the slightly smaller Hagenbach-Bischoff quota, the formula for which is identical to the Droop quota's except that the quotient is not increased to the next whole number. Another difference between the Droop and Hagenbach-Bischoff quotas is that under the Droop quota it is mathematically impossible for more candidates to reach the quota than there are seats to be filled, (although ties are still possible). This can occur under Hagenbach-Bischoff but when it does it is treated as a kind of tie, with one candidate chosen at random for exclusion.