# Single transferable vote

A single transferable vote ballot paper for the electorate of Brindabella in the 2016 Australian Capital Territory general election

The single transferable vote (STV) is a voting system designed to achieve or closely approach proportional representation through the use of multiple-member constituencies and each voter casting a single ballot on which candidates are ranked. The preferential (ranked) balloting allows transfer of votes to produce proportionality, to form consensus behind select candidates and to avoid the waste of votes prevalent under other voting systems.[1] Another name for STV is multi-winner ranked-choice voting.[2]

Under STV, each elector (voter) casts a single vote in a district election that elects multiple winners. Each elector marks their ballot for the most preferred candidate and also marks back-up preferences. A vote goes to the voter's first preference if possible, but if the first preference is eliminated, instead of being thrown away, the vote is transferred to a back-up preference, with the vote being assigned to the voter's second, third, or lower choice if possible (or under some systems being apportioned fractionally to different candidates).

Where there are more candidates than seats, the least popular is eliminated and their votes transferred based on voters' marked back-up preferences. In some systems surplus votes not needed by successful candidates are transferred proportionally, as described below. Elections and/or eliminations, and vote transfers where applicable, continue until enough candidates are declared elected or until there are only as many remaining candidates as there are unfilled seats, at which point the remaining candidates are declared elected.

The specific method of transferring votes varies in different systems (see § Quota and vote transfers).

STV enables votes to be cast for individual candidates rather than for parties or party machine-controlled party lists. Compared to first-past-the-post (FPTP) voting, STV reduces the number of "wasted" votes, which are those cast for unsuccessful candidates and for successful candidates over and above those needed to secure a seat. STV avoids this waste by transferring a vote to another preferred candidate.

STV also provides approximately proportional representation, ensuring that substantial minority factions have some representation. No one party or voting block can take all the seats in a district. The key to STV's achievement of proportionality is that each elector (voter) only casts one single vote, in a district election electing multiple winners.

Under STV, district elections grow more proportionally representative in direct relation to increase in the number of seats to be elected in a constituency – the more seats, the more the distribution of the seats in a district will be proportional. For example, in a three-seat STV election using the Hare Quota of ${\displaystyle ({\rm {\mbox{valid votes cast}}})/({\rm {\mbox{seats to fill}}})}$, a candidate or party needs 33 percent of the votes to win a seat. In a seven-seat STV district, any candidate who can get the support of approximately 14 percent of the vote (either first preferences alone or a combination of first preferences and lower-ranked preferences transferred from other candidates) will win a seat.

Instant runoff voting (IRV) is the single-winner analogue of STV. It is also called single-winner ranked-choice voting. Its goal is representation of a majority of the voters in a district by a single official, as opposed to STV's goal of proportional representation of all the substantial voting blocks by multiple officials.

### Terminology

When STV is used for single-winner elections, it is equivalent to the instant-runoff voting method.[3] STV used for multi-winner elections is sometimes called "proportional representation through the single transferable vote", or PR-STV. "STV" usually refers to the multi-winner version, as it does in this article. In the United States, it is sometimes called choice voting, preferential voting, or preference voting. ("Preferential voting" can also refer to a broader category, ranked voting systems.)

Hare-Clark is the name given to PR-STV elections in Tasmania and the Australian Capital Territory.

### Voting

Simplified example of an STV ballot.

In STV, each voter ranks the candidates in order of preference, marking a '1' beside their most preferred candidate, a '2' beside their second most preferred, and so on as shown in the sample ballot on the right. As noted, this is a simplified example. In practice, the candidates' names are usually organized in columns so that voters are informed of the candidates' party affiliations or whether they are standing as independents. (An alternative way to mark preferences for candidates is to use columns for the voters' preference. One column is to show first preference. An X there goes beside the most preferred candidate. The next column is for the second preference. An X there marks the second-preference candidate., etc.)

### Filling seats under STV

#### The use of quota to fill seats

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 required to win a seat) is derived.[a] Another similarly flexible system set quota at 25 percent of the votes in a district.[4]

Once a quota is determined, candidates' vote tallies are consulted. If a candidate achieves the quota, he or she is declared elected. Then in some STV systems, any surplus vote is transferred to other candidates in proportion to the next back-up preference marked on the ballots received by that candidate. If more candidates than seats remain, the candidate with the fewest votes is eliminated, with their votes being transferred to other candidates as determined by the voters' next back-up preference. Elections and/or eliminations, and vote transfers where applicable, continue until enough candidates are declared elected 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.

It can be shown that a candidate requires a minimum number of votes – the quota (or threshold) – to be elected. A number of different quotas can be used; the most common is the Droop quota, given by the floor function formula:

${\displaystyle {\text{votes needed to win}}=\operatorname {floor} \left({\frac {\text{valid votes cast}}{{\text{seats to fill}}+1}}\right)+1}$

The Droop quota is an extension of requiring a 50% + 1 majority in single-winner elections. For example, at most 3 people can have 25% + 1 in 3-winner elections, 9 can have 10% + 1 in 9-winner elections, and so on.

If fractional votes can be submitted, then the Droop quota may be modified so that the fraction is not rounded down. Major Frank Britton, of the Election Ballot Services at the Electoral Reform Society, observed that the final plus one of the Droop quota is not needed; the exact quota is then simply ${\displaystyle ({\text{valid votes cast}})/({\text{seats to fill}}+1)}$. Without fractional votes, the equivalent integer quota may be written:

${\displaystyle {\text{votes needed to win}}=\operatorname {ceil} \left({{\text{valid votes cast}} \over {\rm {{\text{seats to fill}}+1}}}\right)}$

So, the quota for one seat is fifty out of a hundred votes, not fifty-one.[5]

#### Finding winners using quota

An STV election count starts with a count of each voters' first choice, recording how many for each candidate, calculation of the total number of votes and the quota and then taking the following steps:

1. A candidate who has reached or exceeded the quota is declared elected.
2. 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. This can be done in several ways (see § Quota and vote transfers ).
3. If no-one has exceeded the quota or after all surplus votes have been transferred, the candidate with the fewest votes is eliminated and the votes are transferred to the next preferred candidate marked on each ballot.
4. This process repeats until either every seat has been filled by candidates surpassing quota or until there are as many remaining seats as there are remaining candidates, at which point the remaining candidates are declared elected.

There are variations, such as how to transfer surplus votes from winning candidates and whether to transfer votes to already-elected candidates. When the number of votes transferred from the losing candidate with the fewest votes is too small to change the ordering of remaining candidates, more than one candidate can be eliminated simultaneously.

One simplistic formula for how to transfer surplus votes is: