Editing Single transferable vote (section)

==Seat filling by quota==
{{more citations needed section|date=March 2023}}
{{Main|Counting single transferable votes}}

In most STV elections, a quota is established to ensure that all elected candidates are elected with approximately equal numbers of votes. In some STV varieties, votes are totalled, and a quota (the minimum number of votes that guarantees election) is derived. Those who are elected are the most popular, and attainment of quota is the benchmark of that popularity. Some say that the importance of quota is to set the number of votes that are surplus; that is, the number that should be transferred away from successful candidates.

A common formula sets quota as a fraction of the votes cast. A four-seat district using the Hare quota sets quota as one-fourth of the valid votes; a four-seat district using the Droop quota sets the quota as one more than one-fifth of the valid votes.<ref>Sandford Fleming, Essays on the Rectification of Parliament (1892)</ref>

In some implementations, a "uniform quota" is simply set by law{{snd}}any candidate receiving that set number of votes is declared elected, with surplus transferred away. Something like this system was used in New York City from 1937 to 1947, where seats were allocated to each borough based on voter turnout. Under such a system, the number of representatives elected varied from election to election depending on voter turnout. In the [[1937 New York City Council election]], 26 councillors were elected; in the [[1939 New York City Council election]], newspapers reported that it was expected that the number of councillors would drop to 17 due to lower voter turnout.<ref>{{cite web | url=https://fairvote.org/report/proportion_representation_in_new_york_city_1936_1947/ | title=Proportional Representation in New York City, 1936–1947 }}</ref> Under NYC's STV, total seats on council varied: [[1937 New York City Council election]] 26 seats, [[1939 New York City Council election]] 21 seats, 1941 26 seats, 1943 17 seats, and 1945 23 seats.<ref>{{Cite web | url=https://repository.library.georgetown.edu/bitstream/handle/10822/1044631/Santucci_georgetown_0076D_13763.pdf?sequence=1&isAllowed=y | title=Three Articles on Proportional Representation in American Cities (with an Introduction) | first=Jon M. |last=Santucci | date=15 May 2017}}</ref>

Once a quota is determined, candidates' vote tallies are consulted. If at any time a candidate achieves the quota, they are declared elected. Then if there are still unfilled seats, in some STV systems, any surplus votes (those over and above the quota) are transferred to other candidates in proportion to the next-highest preference marked on all or some of the ballots that had been received by that candidate, if any.

Usually one or more candidates achieve quota in the first count. If there are still unfilled seats after the surplus is transferred, the count would proceed with the candidate with the fewest votes being eliminated. Their votes would be transferred to other candidates as determined by those voters' next preference, if any. Elections and eliminations, and vote transfers where applicable, continue until enough candidates are declared elected to fill the open seats or until there are only as many remaining candidates as there are unfilled seats, at which point the remaining candidates are declared elected. These last candidates may be elected without surpassing quota, but their survival until the end is taken as proof of their general acceptability by the voters.

===Election===
<!-- Please keep this initial explanation simple and put the more complicated nuances in later sections and sub articles -->
An STV election count starts with a count of each voter's first choice, recording how many for each candidate, calculation of the total number of votes and the quota and then taking the following steps:

# A candidate who has reached or exceeded the quota is declared elected.
# If any such elected candidate has more votes than the quota, surplus votes are then transferred to other candidates proportionally based on their next-indicated choice on all the ballots that had been received by that candidate. There are several different ways to do this. (see {{section link||Vote transfers and quota}} ).
# If there are still seats to be filled after the surplus votes of all candidates elected in the first count have been transferred, if any new candidates have been elected, their surplus votes are transferred proportionally.
# If there are still seats to be filled after all surplus votes have been transferred, the candidate with the fewest votes is eliminated and their votes are transferred to the next candidate marked on each ballot. Candidates already elected or eliminated cannot receive votes in most systems.
# This process repeats until either every seat has been filled by candidates surpassing quota or until there are only as many remaining candidates as there are remaining seats, at which point the remaining candidates are declared elected.

There are variations in conducting transfers (see {{section link||Vote transfers and quota}}).

When the number of votes transferred from the losing candidate with the fewest votes is too small to change the ordering of remaining candidates, no transfer is made or more than one candidate is eliminated simultaneously. In most systems, once a candidate has been eliminated or elected, they do not receive any more votes.<ref>Hoag and Hallett, ''Proportional Representation'', p. 88</ref>

===Vote transfers and quota===
{{See also|Comparison of the Hare and Droop quotas}}
STV systems primarily differ in how they transfer surplus votes and in the size of the quota. For this reason, it has been suggested that STV can be considered a family of voting systems rather than a single system.

If fair results are to be produced and the number of candidates is fixed, a quota must be set such that any candidate who receives that many votes is elected. The quota, if used, must be set at a level where no more candidates can reach quota than there are seats to be filled. It cannot be so small that more candidates can be elected than the number of open seats, but the smaller it is, the fairer the result. There are several ways to specify quotas.

The Droop quota is the one most commonly used. It is generally considered to be the absolute lowest number that elects the correct number of candidates to fill the available seats, at least based on the original number of votes cast.

The [[Droop quota]] is given by the [[floor function]] formula:

<math display="block">\text{votes needed to win} = \left\lfloor \frac{\text{valid votes cast}}{\text{seats to fill}+1} \right\rfloor + 1</math>

where <math>\lfloor \ldots \rfloor</math> produces the integer less than or equal to its argument. The Droop quota is an extension of the [[majoritarian]] principle of a successful candidate having to get at least 50% + 1 in single-winner elections. No one else can get as much. In a three-seat contest, 25% plus 1 is the Droop quota because no more than three people can each have 25% of the vote + 1; using Droop means 10% of the vote + 1 is the quota in a nine-seat district because no more than nine people can each have 10% of the vote + 1, and so on.

Droop being relatively low means that the largest party, if it has the majority of votes, is likely to take the majority of the seats in a district. Additionally, a small party may have a chance to take a seat.
	
The [[Hare quota]] was used in the original proposals by [[Thomas Hare (political scientist)|Thomas Hare]].{{sfn|Lambert|Lakeman|1955|p=245}} It is larger than the Droop and sometimes ensures greater representation to less-popular parties within a district. But also, being larger than Droop, Hare presents more of an obstacle to small parties that hope to take just one seat. Being smaller than Hare, the Droop quota may give a seat to a small party that does not have the votes to take a seat under Hare.

Surplus votes cast for a winning candidate are sometimes transferred to the voter's next choice candidate, who is also preferred by the voter. (Any vote is only used once but may be allocated to different candidates along the way until it finds its final place.) Most first-count votes cast for a candidate who wins in the end are never transferred – just the surplus votes are transferred (unless all seats are already filled). Alternate preferences are only consulted if the candidate is unpopular or elected, and not always then. Votes lie where they are when the last seats are filled, so even under STV not all votes are used to elect someone.<ref name="auto">''A Report on Alberta Elections'' (1982)</ref>

There are variations in the conduct of transfers in different variations of STV, such as how to transfer surplus votes from winning candidates and whether to transfer votes to already-elected candidates.

It can happen that a vote is eligible to be transferred but cannot be because it bears no subsequent preference for any remaining candidate. In the case of transfers of surplus votes, an "exhausted" vote remains with the victorious candidates and only transferable votes (votes bearing a usable alternate preference) are used to determine the transfer of the surplus. If the number of transferable votes is less than the number of the surplus, no calculations are needed to make the transfer. Transfer of the transferable votes is done simply by reference to subsequent preference on the votes. Not all the surplus will be transferred if there are not enough transferable votes.

The STV systems in use in government elections today (such as in Malta and Ireland) do not allow votes to be transferred to candidates already elected. If the variation of STV used allows transfers to candidates already elected, when a candidate is eliminated and the next preference on the ballot shows preference for a candidate already elected, votes are transferred to the already victorious candidate, forming a new surplus. The new surplus votes for the victorious candidate (transferred from the eliminated candidate) are then transferred to the next preference of the victorious candidate, as happened with their initial surplus, but just using the recently transferred votes as guide. Vote transfers from the victorious candidate to a candidate who has been eliminated are impossible, and reference must be made to the next marked preference, if any. See {{section link||Seat filling by quota}} for details.

A different quota, one set lower than Droop, is sometimes workable. If fractional votes are used in an STV method, a quota smaller than the Droop quota may be used, where less than a whole number is added to votes/(seats plus 1).

The use of an even smaller quota is sometimes defended, although under such a quota, it is theoretically possible to have more candidates receive quota than the number of empty seats. Frank Britton, of the Election Ballot Services at the Electoral Reform Society, stated that the final "plus one" of the Droop quota is not needed; the quota he proposed was simply <math>(\text{valid votes cast}) / (\text{seats to fill}+1)</math>. The equivalent integer quota may be written:

<math display="block">\text{votes needed to win} = \left\lceil \frac{\text{valid votes cast}}{\rm \text{seats to fill}+1}\right\rceil </math>

So, the quota for one seat is 50 of 100 votes, not 51.{{sfn|Newland|1984}}

Even a low quota, such as the [[Imperiali quota]], is sometimes used. In any case, in most STV elections the appearance of non-transferable votes means that the quota could be lowered significantly below Droop during the counting of the vote with no danger of having too many achieve quota.

In STV, vote transfers are of two types{{Snd}}transfers of votes of eliminated candidates and transfers of surplus votes of elected candidates. The first type happens more often than the second type. Surplus votes are transferred only after a candidate is elected and then only if there are still open seats to be filled and if the transfers may affect the ranking of the remaining candidates, although rules vary from STV system to STV system.

===Transfers of votes of eliminated candidates===
Transfers of votes of eliminated candidates is done simply, without the use of complex math. The next usable preference on the vote gives the destination for the transfer of the vote. If there is no usable preference on the ballot, the vote goes to the "exhausted" or non-transferable pile.

===Transfers of surplus votes===
Various methods are used in STV systems to transfer surplus votes held by elected candidates. The transfer of surplus votes of an elected candidate may be very simply done or may be done more intricately, depending on the circumstances and the choice of the government or election officials.

It can happen that a vote is set to be transferred but cannot be because it bears no subsequent preference for any remaining candidate. In transfers of surplus votes, any non-transferable votes are left with the elected candidate.

If the number of transferable votes is less than the surplus, the transfer of surplus votes can be performed just as it is done in the case of transfer of votes of eliminated candidates, the only difference being that non-transferable votes remain with the elected candidate. They do not go to the exhausted pile. Transfer of the transferable votes is done in these cases simply by reference to the next usable preference on the vote.

In cases where the number of transferable votes is more than the surplus, a more-involved method may be used to make the transfer proportional and to ensure that the quota left with the successful candidate is proportional as well. But election officials here have a choice of using simpler methods or more involved methods.

Votes to the number of the surplus can be drawn at random from the candidate's votes. Choosing the votes at random from the pile means that each transfer should be mixed and will likely closely resemble the composition of the entire pile. (This is the system used in [[Cambridge, Massachusetts]], city elections.) <ref>{{cite web | url=https://www.opavote.com/methods/cambridge-stv-rules | title=Cambridge STV Rules }}</ref>

In the STV systems used in the Republic of Ireland (except Senate elections) and Malta, the next preference is examined and then surplus votes are transferred as whole votes in proportion to the proportions of votes marked for each of the other candidates. This is called the "exact method".<ref>{{cite book |last1=Hoag |first1=Clarence Gilbert |first2=George Hervey |last2=Hallett |title=Proportional representation |location=New York |publisher=The Macmillan Company |date=1926}}</ref> Randomness may arise from the later preferences, if any, if they have to be used later. But if they do have to be used later, choosing the votes at random to compose each transfer means that the votes that make up each transfer should carry back-up preferences in approximately true proportion to the whole.

The basic formula for how to transfer surplus votes when there are more transferable votes than the surplus to be transferred is:

<math display="block">
\begin{align}
& \text{transferred votes given to the next preference} \\[6pt]
= {} & \left( \frac{\text{votes for next preference belonging to the original candidate}} {\text{total votes for the original candidate or total transferable votes}} \right) \times \text{surplus votes for original candidate}
\end{align}
</math>

This can produce fractional votes, which are handled differently under different [[Counting single transferable votes|counting methods]].

Transferring votes without considering later preferences may influence later transfers and such systems are sometimes thought of as being random. Alternatively, some jurisdictions use systems that break down the elected candidate's votes into many separate piles, separating the various combinations of marked preferences on the ballots, or do the same by transferring part of each vote at the transfer value rate. The vote is transferred in the form of the ballot paper, carrying its own back-up preferences with it for possible later use. This is the [[Gregory method]].

The Gregory method (also known as Newland–Britain or Senatorial rules) eliminates randomness by examining all the preferences marked on the last parcel of ballots received by the elected candidate. The later preferences dictate how later transfers, if any, will go. Votes are transferred as fractions of votes. Gregory is in use in Northern Ireland, the Republic of Ireland (Senate elections) and in some electoral systems used in Australia.

Variants exist under the names inclusive Gregory method (IGM) and the weighted inclusive Gregory method (WIGM).{{sfn|Hill|Wichmann|Woodall|1987}}<ref>{{Cite web |last=Gilmour |first=James |date=March 2021 |title=Review of some aspects the Single Transferable Voting system for local elections in Wales |url=https://www.researchgate.net/publication/350495590 |access-date=26 November 2022 |website=ResearchGate}}</ref> WIGM is used in the Scottish local government elections.{{Citation needed|date=January 2024}} Unlike the ordinary Gregory method, these systems look at secondary preferences on all the votes held by the elected candidate, not just the votes that make up the last parcel of votes received.<ref>{{Cite web| title=Making Every Vote Count – The Case for Electoral Reform in British Columbia | url=https://citizensassembly.arts.ubc.ca/resources/TechReport(full).pdf | archive-url=https://web.archive.org/web/20171115220437/http://citizensassembly.arts.ubc.ca:80/resources/TechReport(full).pdf | archive-date=15 November 2017}}</ref>

Both Gregory and earlier methods have the problem that, in some circumstances, they do not treat all votes equally. For this reason, [[Meek's method]], [[Warren's method]] and the [[Wright system]] were invented.

Meek, in 1969,{{sfn|Meek|1994a}} was the first to realize that computers make it possible to count votes in a way that is conceptually simpler and closer to the original concept of STV. One advantage of Meek's method is that the quota is adjusted at each stage of counting when the number of votes decreases because some become non-transferable.<!--Add some history of when Meek came along, please. Note he was attempting to eliminate the problem of tactical voting while still maintaining proportionality and such--> Meek also considered a variant of his system which allows for equal preferences to be expressed.{{sfn|Meek|1994b}} This has subsequently (since 1998) been used by the [[John Muir Trust]] for electing its trustees.<ref>{{cite web |title=Examples of STV elections |url=http://www.macs.hw.ac.uk/~denis/stv_elections |publisher=Heriot-Watt University}}</ref>