# Hare quota

The Hare quota (sometimes called the simple, ideal, or Hamilton quota) is the number of voters represented by each legislator in an idealized system of proportional representation where every vote is used to elect someone. The Hare quota is equal to the number of votes divided by the number of seats.
The Hare quota was used in Thomas Hare's proposal for a single transferable vote system and can still be used for this purpose, though the Droop quota is used for most STV elections today. The Hare quota is often used to set electoral thresholds and to calculate apportionments under party-list proportional representation when using the largest remainder method. In such cases, the Hare quota gives unbiased apportionments that do not favor either large or small parties. However, unlike Droop's quota, the Hare quota does not guarantee a party with a majority of votes in the district will win at least half the seats.
The quota was first proposed by Alexander Hamilton for use in United States congressional apportionment, as part of what is now called Hamilton's method.

## Formula

The Hare quota may be given as:
    ${\frac {{\mbox{total}}\;{\mbox{votes}}}{{\mbox{total}}\;{\mbox{seats}}}}$
where
* Total votes = the total valid poll; that is, the number of valid (unspoilt) votes cast in an election.
* Total seats = the total number of seats to be filled in the election.

## Use in STV

In an STV election a candidate who reaches the quota is elected while any votes a candidate receives above the quota in many cases have the opportunity to be transferred to another candidate in accordance to the voter's next usable marked preference. Thus the quota is used both to determine who is elected and to determine the number of surplus votes when a person is elected with quota. When the Droop quota is used, often about a quota of votes are not used to elect anyone (a much lower proportion that under the first-past-the-post voting system) so the quota is a cue to the number of votes that are used to actually elect someone.
The Hare quota was devised by Thomas Hare, one of the first to work out a complete STV system. In 1868, Henry Richmond Droop (1831–1884) invented the Droop quota as an alternative to the Hare quota. The Hare quota today is rarely used with STV due to fact that Droop is considered more fair to both large parties and small parties.
The number of votes in the quota is determined by the district magnitude of the district in conjunction with the number of valid votes cast.

### Example

To see how the Hare quota works in an STV election, imagine an election in which there are two seats to be filled and three candidates: Andrea, Brad, and Carter. One hundred voters voted, each casting one vote and marking a back-up preference, to be used only in case the first preference candidate is un-electable or elected with surplus. There are 100 ballots showing preferences as follows:
| Number of voters | 60 voters | 26 voters | 14 voters |
| 1st preference | Andrea | Brad | Carter |
| 2nd preference | Carter | Andrea | Andrea |
Because there are 100 voters and 2 seats, the Hare quota is:
    ${\frac {100}{2}}=50$
To begin the count the first preferences cast for each candidate are tallied and are as follows:
* Andrea: 60
* Brad: 26
* Carter: 14
Andrea has reached the quota and is declared elected. She has 10 votes more than the quota so these votes are transferred to Carter, as specified on the ballots. The tallies of the remaining candidates therefore now become:
* Brad: 26
* Carter: 24
At this stage, there are only two candidates remaining and one seat open. The most-popular candidate is declared elected; the other is declared defeated.
Although Brad has not reached the quota, he is declared elected since he has more votes than Carter.
The winners are therefore Andrea and Brad.

## Use in party-list PR

Hong Kong and Brazil use the Hare quota in largest-remainder systems.
In Brazil's largest remainder system the Hare quota is used to set the basic number of seats allocated to each party or coalition. Any 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.

### In Hong Kong

For geographical constituencies, the SAR government adopted weakly-proportional representation using the largest remainder method with Hare quota in 1997. Typically, largest remainders paired with the Hare quota produces unbiased results that are difficult to manipulate. However, the combination of extremely small districts, no electoral thresholds, and low led to a system that parties could manipulate using careful vote management.
By running candidates on separate tickets, Hong Kong parties aimed to ensure they received no seats in the first step of apportionment, but still received enough votes to take several of the remainder seats when running against a divided opposition. 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 has effectively created a multi-member single-vote system in the territory.

## Mathematical properties

In situations where parties' total share of the vote varies randomly, the Hare quota is the unique unbiased quota for an electoral system based on vote-transfers or quotas. However, if the quota is used in small constituencies with no electoral threshold, it is possible to manipulate the system by running several candidates on separate lists, allowing each to win a remainder seat with less than a full quota. This can make the method behave like the single non-transferable vote in practice, as has happened in Hong Kong. By contrast, rules based on the Droop quota cannot be manipulated in the same way, as it is never possible for a party to gain seats by splitting.