A quota is a number of votes (obtained by formula) often relevant to deciding who wins and how ballots are evaluated or modified in "[[ PR|proportional representation]]" voting methods]]. ▼{{mbox|text=I would like to break this page up into three or four pages (minimally: "Quota", [[Hare quota]] and [[Droop quota]]). I'm likely to [[Project:Be bold|be bold]] and just do it, but please leave comments on [[Talk:Quota]] if you object. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 00:44, 5 December 2020 (UTC)}}
The two main quotas that will be described here are the "[[Hare quota]]" and the "[[Droop 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 [[PR|proportional voting methods]].
The following quotas are listed from largest to smallest. See the [[PSC#Types of PSC]] article for more information.
== Hare quota ==
{{wikipediamain|Hare quota}}
The Hare quota may be given as:
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 system]]s 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.
:<math>\frac{\mbox{total} \; \mbox{votes}}{\mbox{total} \; \mbox{seats}}</math>
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.
Where:
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.
*<math>\text{total votes}</math> = the total valid poll; that is, the number of valid (unspoiled) votes cast in an election.
*<math>\text{total seats}</math> = 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.
== Droop quota ==
{{wikipediamain|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]].
Sources differ as to the exact formula for the Droop quota. As used in the Republic of Ireland the formula is usually written:
<math >\left( \frac{\text{total valid poll}}{ \text{seats}+1 } \right) + 1</math>
but more precisely
<math >\operatorname{Integer} \left( \frac{\text{total valid poll}}{ \text{seats}+1 } \right) + 1</math>
where:
* <math>\text{total valid poll}</math> = Total number of valid (unspoiled) votes cast in an election.
* <math>\text{seats}</math> = total number of seats to be filled in the election.
* <math>\operatorname{Integer}()</math> refers to the integer portion of the number, sometimes written as <math>\operatorname{floor}()</math>
One reason Droop quotas are used more often than Hare Quotas for ranked [[Proportional representation|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.
== Hagenbach-Bischoff quota ==
{{wikipedia|Hagenbach-Bischoff quota}}
The '''Hagenbach-Bischoff quota''' ('''HB quota''') (known by a few other names as well) is:
<math>\left( \frac{\text{total valid poll}}{ \text{seats}+1 } \right)</math>
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.
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'.
When there are 5 seats to be filled and 100 votes cast, the HB quota is (100/(5+1)) = '''~16.667''' votes.
== References ==
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).
<references/>
[[Category:Proportionality-related concepts]]
[[Category:Quotas|*]]
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