# Droop Quota

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The Droop Quota is the formula that is used to calculate the minimum number, or quota, of votes required to capture a seat in a multi-member constituency using Proportional Representation through the Single Transferable Vote (PR.STV). It was devised in 1868 by the English lawyer and mathematician Henry Richmond Droop (1831-1884). A similar formula was developed by Eduard Hagenbach-Bischoff from Switzerland for Party-List Proportional Representation.

It is used in all elections in the Republic of Ireland, Malta and elections to the Tasmanian House of Assembly, among others. Sources differ as to the exact formula for the Droop Quota, and also on whether it is a proper noun.

The formula used in the Republic of Ireland is usually written:

[itex]\left( \frac{\mathrm{Total\ Valid\ Poll}}{\left( \mathrm{Seats}+1 \right)} \right) + 1[itex]

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 crucially important when calculating to use the Total Valid Poll (TVP), which is arrived at by subtracting the spoiled and invalid votes from the total poll. When calculated correctly, 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 strict mathematical point of view, the formula may best be rendered:

The brackets denote the operation of rounding down; i.e. the Droop Quota is the smallest integer greater than Votes / (Seats + 1).

This gives the Droop Quota the special property that it is the smallest integral quota (although not the smallest quota) which guarantees that the number of candidates able to reach this quota cannot exceed the number of seats. In the case of a single seat, it, of course, degenerates into a simple integral majority quota (Instant-runoff voting).

## Example

If there are 100,000 votes cast in a constituency that has 5 seats, that produces (100,000/6) + 1 = 16,667. So the quota each candidate needs to reach is 16,667. Using PR.STV, each voter on a ballot paper receives a list of candidates, with the option of listing their preferences (ie, how high they rate in the choice of the voter to win a seat). So the favourite candidate receives a number 1, the second favourite number 2, third favourite number 3, etc. Each voter has absolute freedom to vote for as many or as few candidates they wish. In the first count, all the number 1s are calculated. Where someone exceeds the quota, their surplus (total vote minus quota) is calculated. That surplus is analysed to see which candidates were the number 2 choice of those votes the candidate had over and above the quota, which was the minimum number of votes needed for election. If no candidate reached the quota, the candidate with the lowest vote total would be eliminated, and their votes distributed. A series of counts would take place, in which either a surplus or elimination total would be distributed. In theory, enough candidates should reach the quota, leaving not enough votes left distributed among the remaining candidates for a remaining sixth quota. Once the last candidate is at an end, the election count is declared over.

While in theory every election should see the right number of candidates elected through reaching the quota, in practice a minority of voters may only vote for a small proportion of the candidates on a ballot paper, for example only those candidates from one party, candidates of one gender, or even simply 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 may not actually 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".

In summary the quota is constructed to ensure that it is mathematically impossible for anyone other than the five candidates elected to reach the quota.

A more detailed example is at: Single Transferable Vote

• 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 (1881) pp.141-196 [Discussion, 197-202].et:Droopi kvoot

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