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In the study of apportionment, the Hare quota (sometimes called the simple, ideal, or Hamilton quota) is the number of voters represented by each legislator under an idealized system of proportional representation, where every legislator represents an equal number of voters and where every vote is used to elect someone. The Hare quota is the total number of votes divided by the number of seats to be filled. The Hare quota was used in the original proposal for a single transferable vote system, and is still occasionally used, although it has since been largely supplanted by the Droop quota.

The quota continues to be used in setting electoral thresholds, as well as for calculating apportionments by the largest remainder method (LR-Hare) or other quota-based methods of proportional representation. In such cases, the Hare quota gives unbiased apportionments that favor neither large nor small parties.^[1] However in certain circumstances when the Hare quota is used, a party with a majority of votes in the district would not win a majority of the seats.^[2]

In 1792, a U.S. national figure, Alexander Hamilton, proposed to use the Hare quota to establish representation by population, by fairly allocating seats in the House of Representatives to each state, with the largest remainder method used to allocate seats to states with remaining leftover partial quota units of population.[1][2]

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.^[3]

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.^[4]

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.^[5] 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^{[citation needed]}. Typically, largest remainders paired with the Hare quota produces unbiased results that are difficult to manipulate.^[1] 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.^[6] 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.^[7]^[8]^[9]

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.^[1] 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.^[9] By contrast, Droop quota cannot be manipulated in the same way, as it is never possible for a party to gain seats by splitting.^[1]

References

^ ^a ^b ^c ^d Pukelsheim, Friedrich (2017). "17". Proportional Representation. SpringerLink. pp. 108–109. doi:10.1007/978-3-319-64707-4. ISBN 978-3-319-64707-4.
^ Humphreys, Proportional Representation (1911), p. 138
^ Baily, PR in large constituencies (1872) (hathitrust online)
^ Baily, PR in large constituencies (1872) (hathitrust online)
^ (in Portuguese) Brazilian Electoral Code, (Law 4737/1965), Articles 106 to 109.
^ Tsang, Jasper Yok Sing (11 March 2008). "Divide then conquer". South China Morning Post. Hong Kong. p. A17.
^ Ma Ngok (25 July 2008). 港式比例代表制議會四分五裂 [Hong Kong-style proportional representation is divided]. Ming Pao (in Chinese (Hong Kong)). Hong Kong. p. A31.
^ Choy, Ivan Chi Keung (31 July 2008). 港式選舉淪為變相多議席單票制 [Hong Kong-style elections become a multi-seat multi-seat single-vote system]. Ming Pao (in Chinese (Hong Kong)). Hong Kong. p. A29.
^ ^a ^b Carey, John M. "Electoral Formula and Fragmentation in Hong Kong" (PDF). {{cite journal}}: Cite journal requires |journal= (help)