Single transferable vote

The Single Transferable Vote (STV) is an electoral system that yields proportional representation. In a Single Transferable Vote election, voters rank candidates or parties in order of preference, and candidates are elected sequentially. Proportionality is maintained by associating a cost, or quota, with the election of each candidate, which a candidate's supporters must pay when that candidate is elected.

In a single winner election, it is Instant Runoff Voting.

Procedure
While there are several variants of STV, most implementations of it more or less work as described below.

Finding the Quota
First, the number of valid votes cast is used to compute the quota, that is, the number of votes that a candidate needs to be elected. The two major quotas are the Hare Quota, which is equal to Votes/Seats and the Droop Quota, which is equal to Votes/(Seats+1), rounded up. While the Hare Quota makes it possible for every vote to be assigned to a candidate, a candidate only needs to exceed the Droop Quota to guarantee that they will be elected. For this reason, using the Hare Quota can make a method more vulnerable to strategic voting, so the Droop Quota is more commonly used in practice.

Counting the Votes
Initially, each vote starts out as having a weight of 1, and is assigned to the candidate that is ranked first on the ballot. A candidate's vote count is the sum of the weights of the votes assigned to them.

Any candidates with vote counts that are at least as large as the quota are declared elected. If an elected candidate has more votes than the quota, the amount by which their vote count exceeded the quota is called a surplus, and the ballots supporting the elected candidate have their weight multiplied by the ratio between the surplus and the total votes assigned to the elected candidate. (If there is no surplus, the weight is reduced to 0.) Then, the votes are transferred to the highest ranked candidate on the ballot who has not been elected or eliminated. If no such candidates are ranked on the ballot, the ballot is considered exhausted, and is ignored for the remainder of the election.

If no candidates exceed the quota, then the candidate with the smallest vote total is eliminated, and their votes are transferred in the same way as they would be for surplus transfers, except that the votes do not lose any weight.

The counting stops when all the seats are filled, or as many candidates as unfilled seats remain, in which case they are filled by the remaining candidates.

Example
Consider the following 3 winner election with the ballots There are 36 votes and 3 seats, so the Droop quota is 36/(3+1)=9 votes.

The first count is Alice: 21, Aaron: 0, Bob: 4, Brenda: 4, Charlie: 7. Only Alice exceeds the quota, so she is elected and each vote for Alice has its weight multiplied by (12/21). All of her votes transfer to Aaron.

The second count is Aaron: 12, Bob: 4, Brenda: 4, Charlie: 7. Only Aaron exceeds the quota, so he is elected, and each vote for him has its weight multiplied by (3/12). The Alice>Aaron votes are exhausted as all of the candidates they listed were elected. The Alice>Aaron>Bob votes transfer to Bob.

The third count is Bob: 4.952, Brenda: 4, Charlie: 7. No candidate exceeds the quota, so the one with the fewest votes, Brenda is eliminated. Her votes transfer to Bob.

The fourth count is Bob: 8.952, Charlie: 7, so Charlie is eliminated and Bob wins the third seat.

Use in National Elections
STV is used to elect the Australian Senate, Irish Dáil Éireann, and Maltese House of Representatives. Ireland and Malta also use it for all local and regional elections as well.

Use in Local Elections
Some Australian state and territorial houses are elected with STV.

STV is used in local and regional elections in Northern Ireland except for MPs of the UK House of Commons. It is used for local elections in Scotland.

It was used in municipal elections in the Canadian provinces of Alberta and Manitoba between the mid-1920s to mid-1950s.

In the US, it is currently used in Cambridge, Massachusetts. , and Minneapolis, Minnesota. It was used in several major American cities such as New York during the first half of the 20th century, but during the 1950s it was largely abandoned because it allowed racial minorities and Communists to be elected.

Benefits
STV can provide proportional representation while still allowing voters to vote for individual candidates instead of political parties, unlike party list systems, where voting for a party means supporting the entire list. It also makes it easier for independent candidates to get elected compared to other proportional representation systems.

Transferring votes helps mitigate wasted votes, in contrast with list PR systems, especially those with thresholds, which effectively discard votes for parties below the threshold.

Difficulty of judging individual candidates
Even when voters can vote for individual candidates (and not just parties), it is more difficult than party voting, so voters will often prefer to vote for parties if given the option. For example, in Australian Senate elections, very few voters vote "below the line".

Furthermore, some regional elections in Australia allow minor parties to direct the preferences of above-the-line voters when they are eliminated if the voter ranks only one party; this rule allows minor parties to make deals that can allow obscure candidates to be elected with the votes of voters who may not have even heard of the candidate whom their votes elected.

Free Riding
Free riding occurs in STV elections when a voter attempts to avoid contributing to the election of a candidate whom they are sure will be elected, in order to avoid paying the cost of electing that candidate. There are two major ways of doing this.

Woodall Free Riding
Woodall Free Riding occurs when a voter intentionally gives their first preference to a candidate they expect to lose quickly at the top of the ballot. This ensures that they will not have to pay for the election of a candidate until after the first eliminations, since that candidate will have to be eliminated in order for their vote to be transferred. In particular they will not have to pay for the election of a candidate who exceeds the quota on the first count.

Meek STV, used in RationalWiki moderator elections, allows votes to be transferred to already-elected candidates in order to prevent Woodall Free Riding. So don't get any big ideas!

Hylland Free Riding
Hylland Free Riding occurs when a voter does not rank any candidates whom they are sure will be elected. When it is practiced en masse, it leads to several candidates being elected with less than a full quota, magnifying the power of the votes for that candidate.

Vote Management
Vote management involves coordinated campaigns of Hylland Free Riding, typically by a political party. The strategy is for the party to estimate the number of seats that they believe they will be able to compete for, dividing their voters into groups of roughly equal size, and assigning each group to one of that party's candidates, whom they are instructed to give their first choice to. The advantage of this tactic is demonstrated in the following example.

As there are 108 votes, the Droop quota is 108/6=18.

Candidate B1 will be elected first, followed by B2. This will cost the B voters 36 votes, leaving them with 15 votes worth of weight.

After that, no candidate will have exceeded the quota, so C will be eliminated, followed by B3, leaving A1, A2, and A3 as the winners of the remaining seats. This allows party A to win a 3-2 majority of the seats despite being outnumbered 51-48 by party B voters.

It is commonly practiced in Irish elections. A consequence of vote management in the 2020 Irish election was that Sinn Féin won slightly fewer seats than their proportional share of the vote, since they received a late increase in support too late to run more candidates and thus underestimated the number of seats they could win in some constituencies.

Monotonicity and Spoilers
As a multiwinner analogue of Instant Runoff Voting, STV inherits some of its flaws as well, such as nonmonotonicity and participation failure.