Droop Quota
Encyclopedia
The Droop quota is the quota most commonly used in elections held under the Single Transferable Vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

 (STV) system. It is also sometimes used in elections held under the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...

 of party-list proportional representation
Party-list proportional representation
Party-list proportional representation systems are a family of voting systems emphasizing proportional representation in elections in which multiple candidates are elected...

 (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
Henry Richmond Droop
Henry Richmond Droop * was an English mathematician. He is widely credited with devising the Droop quota that is used today in the common voting system known as the Single Transferable Vote. He also may have been the first to write down what later became known as Duverger's Law, in 1869.He married...

 (1831–1884) as a replacement for the earlier Hare quota
Hare quota
The Hare quota is a formula used under some forms of the Single Transferable Vote system and the largest remainder method of party-list proportional representation...

.

Today the Droop quota is used in almost all STV elections, including the forms of STV used in the Republic of Ireland
Republic of Ireland
Ireland , described as the Republic of Ireland , is a sovereign state in Europe occupying approximately five-sixths of the island of the same name. Its capital is Dublin. Ireland, which had a population of 4.58 million in 2011, is a constitutional republic governed as a parliamentary democracy,...

, Northern Ireland
Northern Ireland
Northern Ireland is one of the four countries of the United Kingdom. Situated in the north-east of the island of Ireland, it shares a border with the Republic of Ireland to the south and west...

, Malta
Malta
Malta , officially known as the Republic of Malta , is a Southern European country consisting of an archipelago situated in the centre of the Mediterranean, south of Sicily, east of Tunisia and north of Libya, with Gibraltar to the west and Alexandria to the east.Malta covers just over in...

 and Australia
Australia
Australia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...

, among other places. The Droop quota is very similar to the simpler Hagenbach-Bischoff quota
Hagenbach-Bischoff quota
The Hagenbach-Bischoff quota is a formula used in some voting systems based on proportional representation . It is used in some elections held under the largest remainder method of party-list proportional representation as well as in a variant of the D'Hondt method known as the Hagenbach-Bischoff...

, 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:
  • 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
Spoilt vote
'Bold text'In voting, a ballot is considered to be spoilt, spoiled, void, null, informal or stray if it is regarded by the election authorities to be invalid and thus not included in the tally during vote counting. This may be done accidentally or deliberately...

 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. Simply put, the Droop quota is the number without its "decimal part". 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
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

, 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
  1. Andrea
  2. Carter

25 voters
  1. Carter

30 voters
  1. 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:


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
  • Brad: 30


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
  • Brad: 30


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

Comparison with the Hare quota

The Droop quota is smaller than the Hare quota
Hare quota
The Hare quota is a formula used under some forms of the Single Transferable Vote system and the largest remainder method of party-list proportional representation...

, and is more efficient when counting ballots since a candidate needs only the smaller quota to be considered elected. Overall the two quotas achieve very similar net results since a candidate cannot not be elected once they achieve the Droop quota, however the results may differ, particularly for the last seat, based on the transfer of preferences.
  • In a multi-winner election, the Hare quota is kinder to small parties than the Droop quota because they have a slightly better chance to win the final seat. The principle of proportional representation favours the Hare quota;
  • In an open list
    Open list
    Open list describes any variant of party-list proportional representation where voters have at least some influence on the order in which a party's candidates are elected...

     multi-winner election under the Hare quota it is possible for a party supported by a clear majority of voters to receive only a minority of seats if the votes are not dispersed relatively evenly across all the party's candidates. The principle of majority rule favours the Droop quota;
  • In an STV election in which there is only one seat to be filled (in other words an Instant Run-off Voting
    Instant-runoff voting
    Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

     election) both quotas will achieve the same result.


The difference between the two quotas comes down to what the quota implies. Winners elected under a Hare system represent that proportion of the electorate; winners under a Droop system were elected by that proportion of the electorate.

The Droop quota is today the most popular quota for STV elections.

Comparison with 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
Hagenbach-Bischoff quota
The Hagenbach-Bischoff quota is a formula used in some voting systems based on proportional representation . It is used in some elections held under the largest remainder method of party-list proportional representation as well as in a variant of the D'Hondt method known as the Hagenbach-Bischoff...

, 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.

Further reading

  • Henry Richmond Droop, On the Political and Social Effects of Different Methods of Electing Representatives. (London, 1869)
  • Henry Richmond Droop, "On methods of electing representatives" in the Journal of the Statistical Society of London Vol. 44 No. 2 (June 1881) pp. 141–196 [Discussion, 197-202], reprinted in Voting matters
    Voting matters
    Voting matters is a peer-reviewed academic journal whose purpose is "To advance the understanding of preferential voting systems". Originally published by the Electoral Reform Society , Voting matters is now a publication of the McDougall Trust...

    Issue 24
    (October 2007) pp. 7–46.
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