Quota

A quota is a number of votes (obtained by formula) often relevant to deciding who wins and how ballots are evaluated or modified in proportional voting methods.

The following quotas are listed from largest to smallest. See the PSC article for more information.

Hare quota
The Hare quota may be given as:


 * $$\frac{\mbox{total} \; \mbox{votes}}{\mbox{total} \; \mbox{seats}}$$

Where:


 * $$\text{total votes}$$ = the total valid poll; that is, the number of valid (unspoiled) votes cast in an election.
 * $$\text{total seats}$$ = the total number of seats to be filled in the election.

When there are 5 seats to be filled and 100 votes cast, the Hare quota is (100/5) = 20 votes.

In the single-winner case, a Hare quota is just all of the voters. In general, voting methods that are based on Hare quotas attempt to represent all voters, but don't guarantee that a majority of voters will get even half of the seats.

English Wikipedia description
Here's how English Wikipedia describes the Hare quota: The Hare quota (also known as the simple quota) is a formula used under some forms of the Single Transferable Vote (STV) system and the largest remainder method of party-list proportional representation. In these voting systems the quota is the minimum number of votes required for a party or candidate to capture a seat, and the Hare quota is the total number of votes divided by the number of seats.

The Hare quota is the simplest quota that can be used in elections held under the STV system. In an STV election a candidate who reaches the quota is elected while any votes a candidate receives above the quota are transferred to another candidate.

The Hare quota was devised by Thomas Hare, one of the earliest supporters of STV. In 1868, Henry Richmond Droop (1831–1884) invented the Droop quota as an alternative to the Hare quota, and Droop is now widely used, the Hare quota today being rarely used with STV.

In Brazil's largest remainder system the Hare quota is used to set the minimum number of seats allocated to each party or coalition. Remaining seats are allocated according to the D'Hondt method. This procedure is used for the Federal Chamber of Deputies, State Assemblies, Municipal and Federal District Chambers.

Compared to some similar methods, the use of the Hare quota with the largest remainder method tends to favour the smaller parties at the expense of the larger ones. Thus in Hong Kong the use of the Hare quota has prompted political parties to nominate their candidates on separate tickets, as under this system this may increase the number of seats they obtain. The Democratic Party, for example, filled three separate tickets in the 8-seat New Territories West constituency in the 2008 Legislative Council elections. In the 2012 election, no candidate list won more than one seat in any of the six PR constituencies (a total of 40 seats). In Hong Kong the Hare quota system has effectively become a multi-member single-vote system in the territory. This formula also rewards political alliances and parties of small-to-moderate size and discourages broader unions which led to the fragmentation of the political parties and electoral alliances rather than expanding them.

The article above may have changed since this writing in December 2020. See Hare quota for the latest version.

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

but more precisely

where:


 * $$\text{total valid poll}$$ = Total number of valid (unspoiled) votes cast in an election.
 * $$\text{seats}$$ = total number of seats to be filled in the election.
 * $$\operatorname{Integer}$$ refers to the integer portion of the number, sometimes written as $$\operatorname{floor}$$

One reason Droop quotas are used more often than Hare Quotas for ranked PR methods is because not only do they often help reduce the amount of vote-counting necessary, but they almost entirely eliminate the possibility of a majority of voters receiving a minority of seats compared to Hare Quotas. The Droop Quota is the smallest possible quota that guarantees that there will be as many quotas as there are winners desired.

When there are 5 seats to be filled and 100 votes cast, the Droop quota is 17 votes, which is calculated as: Integer((100/(5+1)) + 1) = Integer((100/6) + 1) = Integer(~16.667 + 1) = Integer(~17.667) = 17 votes.

In the single-winner case, a Droop quota is a majority. In general, Droop quota-based methods tend to leave at least just under a Droop quota unrepresented. See the utility article, as the debate between Hare and Droop quotas somewhat parallels and generalizes the utilitarianism vs. majority rule debate.

English Wikipedia description of Droop
Below is a description copied from English Wikipedia describing the Droop quota:

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 India, the Republic of Ireland, Northern Ireland, Malta and Australia, among other places. The Droop quota is very similar to the simpler Hagenbach-Bischoff quota, which is also sometimes loosely referred to as the 'Droop quota'.

The article above may have changed since this writing in December 2020. See Droop quota for the latest version.

Hagenbach-Bischoff quota
The Hagenbach-Bischoff quota (HB quota) (known by a few other names as well) is:

$$\left( \frac{\text{total valid poll}}{ \text{seats}+1 } \right)$$

Some sources call the HB Quota a Droop Quota instead. There will always be exactly one more HB quota than seats to be filled. Because of this, it will on rare occasion be necessary to break a tie between various candidates to decide who should win with PR methods that use the HB quota.

When there are 5 seats to be filled and 100 votes cast, the HB quota is (100/(5+1)) = ~16.667 votes.

In the single-winner case, an HB quota is half of the voters. In this case, two candidates could each have half of the votes, i.e. two candidates each have one quota, but only one seat can be allotted. Because of this, many PR methods that use HB quotas specify that a candidate must have  more votes than k HB quotas to get k seats (i.e. over half of the votes, in the single-winner case).