Hare quota

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. 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 use cases, the Hare quota gives unbiased apportionments that favor neither large nor small parties.

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, the first to identify of STV. In 1868, Henry Richmond Droop (1831–1884) invented the Droop quota as an alternative to the Hare quota. The he 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:

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 and 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 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.

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. When the number of seats in each constituency is large and vote management becomes impractical (as it requires an unrealistic degree of coordination of voters), the quota becomes fully unbiased.

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 transform the method into a de facto single nontransferable vote system, a situation that has been seen in Hong Kong. By contrast, Droop's quota cannot be manipulated in the same way, as it is never possible for a party to gain seats by splitting. <!--

In Hong Kong
In the 1998 Hong Kong legislative election, pro-democracy camp organization The Frontier fails to co-ordinate two former legislators (1995–1997) Lee Cheuk-yan and Leung Yiu-chung into a two-candidate list running for New Territories West (NT West) 5-seat constituency, and Leung left The Frontier, running as Nonpartisan candidate with the support of Neighbourhood and Worker's Service Centre in NT West and Lee running as Frontier candidate in NT West. Lee and Leung won the last two seats by around 10% votes (Lee 12.45% and Leung 10.30%), in case they ran in a single list with same election result(12.45% + 10.30% = 22.75%), they would win the first seat by full quota (20% as a 5-seat constituency) and the remainder(2.75%) is smaller than the candidate list standing for indigenous inhabitants of the New Territories, which led by vice-chairman of the Heung Yee Kuk - Lam Wai-keung (6.91%).

In 2000 Hong Kong legislative election, the second legislative election using the Hare quota largest remainder method, fragmentation and infighting within the parties and camps were shown because political parties began to split their lists in order to waste fewer votes as acquiring seats with remainder votes can be more efficient than purchasing them with full quotas under the Hare quota. For instance, the Democratic Party ran multiple lists by filling two lists in New Territories East and three lists in New Territories West, in which incumbent Lee Wing-tat's list was lost to his party colleague Albert Chan's list in the latter constituency. In 2004, the ADPL joined the Democrats by splitting lists in Kowloon West.

In 2012, the pro-Beijing DAB deployed multiple lists for the first time. As a result, of the 34 seats captured by lists from the two major camps, only three were won by full quota. Due to strong network of pro-Beijing camp with its affiliated grassroots and community organisations, pro-Beijing camp was able to split the votes evenly to get more candidates to be elected with fewer votes, pro-Beijing camp won the last seats in 4 out of 5 constituencies and total 17 of 35 geographical Constituency seats with 42.66% shares of votes, compared with pan-democrats 56.24% shares of votes winning 18 seats.

The Hare quota used in Hong Kong's largest remainder system also encourages the multiplication of political parties and nonpartisan candidates. The vote share of the largest party Democratic Party dropped significantly, from 43 per cent in 1998 to 29 per cent in 2000, to 21 per cent in 2004, rising slightly to 20 per cent in 2008 and falling again to 14 per cent in 2012. As it is only possible to win a single remainder seat, politicians and potential allies can be motivated to diverge rather than to coalesce. -->