Single transferable vote

Under STV, multiple winners are selected for a constituency (a multi-member district). Every sizeable group within the district wins at least one seat: the more seats the district has, the smaller the size of the group needed to elect a member. In this way, STV provides approximately proportional representation, ensuring that substantial minority factions have some representation.

On their ballot, the voter ranks candidates in order of preference. A vote is initially allocated to the voter's first preference. If seats remain open after this first count, votes are transferred as per following steps.

If that candidate is eliminated, the vote is transferred to the next-preferred candidate rather than being discarded; if the second choice is eliminated, the procedure is iterated to lower-ranked candidates. Under some systems, the vote is apportioned fractionally to different candidates. As long as there are more candidates than seats, the least popular candidate is eliminated, and votes for them are transferred based on voters' subsequent preferences.

Before the election, a quota (the minimum number of votes that guarantees election) is calculated by a specified method, and candidates who accumulate that many votes are declared elected. In some systems, the quota is also used to determine surplus votes, the amount of votes received by successful candidates over and above the quota. Surplus votes are transferred to candidates ranked lower in the voters’ preferences, so they would not be wasted by remaining with a candidate who does not need them.

Counting, eliminations, and vote transfers continue until enough candidates are declared elected (all seats are filled by candidates reaching the quota) 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 § Vote transfers and quota).

Because of this quota-based fairness, under STV it is extremely rare for a party to take a majority of the seats in a district without a majority of the district vote. Additionally, a large majority of voters (generally around 80 percent or more) see their vote used to elect someone. Thus under STV, the candidates who make up a majority of the district's elected members are supported directly by a majority of the voters in the district.

Suppose an election is conducted to determine what three foods to serve at a party. There are seven choices: Oranges, Pears, Strawberries, Cake (Strawberry-chocolate), Chocolate, Hamburgers and Chicken. Only three of these may be served.

There are 23 guests, and the hope is that each guest will be served at least one food that they are happy with. It is decided to use STV to make the decision. Each guest is given one vote but is also allowed to cast two optional alternate preferences to be used only if the first preference cannot select a food or to direct transfer of surplus votes if it does. The 23 guests at the party mark their ballots: some mark first, second and third preferences; some mark fewer preferences. When the ballots are counted, it is found that the ballots are marked in seven distinct combinations, as shown in the table below:

The table is read as columns: the left-most column shows that there were four ballots with Orange as the first choice, and Pear as second; while the rightmost column shows there were three ballots with Chicken as first choice and Hamburger second.

Step 1: First-preference votes are counted. Pears reaches the quota with 7 votes, and is therefore elected on the first count, with 1 surplus vote

Step 2: All of the voters who gave first preference to Pears preferred Strawberry next, so the surplus vote is awarded to Strawberry. No other option has reached the quota, and there are still two to elect with six options in the race, so elimination of lower-scoring options will start on the next round.

Step 3: Chocolate has the least votes and is eliminated. According to their only voter's next preference, this vote is transferred to Cake. No option has reached the quota, and there are still two to elect with five in the race, so elimination of options will continue next round.

Step 4: Of the remaining options, Strawberry now has the least votes and is eliminated. In accordance to the preferences of both the only voter who voted Strawberry, and the Pear–Strawberry–Cake vote, these votes are transferred to Cake. Cake reaches the quota and is elected. No other option has reached the quota, and there is still one to elect with three in the race, so elimination of options will continue next round.

Step 5: Chicken has the least votes and is eliminated. According to the Chicken voters' next preference, this vote is transferred to Hamburgers. Hamburgers is thus elected with 7 votes in total. Hamburgers now also has a surplus vote, but this does not matter, since the election is over. There are no more foods needing to be chosen – three have been chosen.

STV in this case produced a higher number of effective votes – 19 votes were used to elect the successful candidates. (Only the votes placed for Oranges were neither used to select a food nor transferred.) As well, there was general satisfaction with the choices selected – 14 voters saw their first preference chosen, and the 9 others saw their second preference chosen. In addition, seven saw their first and third choices selected; one saw his second and third choice selected.

Note that if Hamburger had received only one vote when Chicken was eliminated, it still would have won because the only other remaining candidate, Oranges, has fewer votes so would have been eliminated in the next round. This would have left Hamburger as the last remaining candidate to fill the last open seat, even if it did not have quota.

Single non-transferable vote results would have had Orange among the three winners, as opposed to Cake, for having a greater number of first-preference votes. Under SNTV, 15 voters would have seen their first preference win (Oranges, Pears and Hamburgers); only 3 voters would have not seen their first preference food served but would have seen their 2nd preference food served. Five voters would not be served any of their favourites.

Under first-past-the-post, the guests would have been split into three groups with one food chosen by each group based on just the most popular food in each group. The result in this case would have been dependent on how the groups are formed (gerrymandering of the groups to bias the election toward a particular result could also occur). It might have been Strawberry donuts, Pears and Hamburgers, but also the foods chosen might have been Pears in two groups (districts) and Hamburgers in the other. Or even just Pears alone might have won in each of the three "districts", in which case only 7 guests out of 23 would have seen their first choice served, a very unrepresentative outcome, given that three different foods could have been served.

It could happen under any three-district single-winner system that none of the groups elect Pears, if the 7 votes for it are split and in each "district" there is another food that beats it (e.g. Oranges, Hamburgers and Chicken).

If the voters had been able to choose only one food to serve (as in first-past-the-post, but without "districts"), it is likely that Pears, the choice of less than a third of the 23 party-goers, would have won, meaning Pears would be the only food served at the party. Even if they held two rounds of voting, the bare majority that prefers some kind of fruit (Oranges, Pears, Strawberries) would have dominated all other choices.

Elections with parties are conducted in very similar manner to the non-partisan STV election presented above. Parties actually play no role in STV elections – each voter marks preferences for individual candidates and his or her secondary preferences may cross party lines if so desired.

This example shows election of five members in a district. Party A runs five candidates, Party B runs three, and there is one independent in the race. The election is conducted under STV with the Hare quota, which for five candidates is 20% (100% divided by five).

In the first round, the vote tally of the most popular candidate of Party A, Candidate A3, is more than quota, so they win a seat.

Surplus votes are distributed; the voters of Candidate A3 have put another politician from their party, Candidate A4, as their second preference, so A4 now receives Candidate A3's surplus votes. This transfer of 5 percent of the votes leaves A3 with the quota (20 percent) and leaves A4 with 13 percent.

In the third and fourth rounds, the least popular candidates are eliminated (Candidates A1 and A5) and their votes transferred to their next preferences. Voters of Candidate A5 are not very partisan, they actually prefer the independent candidate over the other candidates of Party A still in the race.

In the fifth round, Candidate A2 is eliminated with all their votes going to the candidate A4, the last remaining candidate from Party A, who is elected. The surplus votes of Candidate A4 are transferred. All the voters who helped elect Candidate A4 prefer the independent candidate to the candidates of the other party so their 3% surplus votes will go to Candidate I in the sixth round.

There are now only four candidates remaining and three seats remaining open. The least popular candidate (Candidate B1) is eliminated. There are now only three candidates in the race, so they are automatically declared elected regardless of whether they reached the quota. If there is no reason to establish relative popularity of the elected members, the count ends there when the last seats are declared filled.

If the ranking of the candidates is important, the votes belonging to Candidate B1 might be transferred as per below, assuming voters' alternate preferences are marked that way.

This vote count varies from the reality of many STV systems because there were no "exhausted" non-transferable votes. In most real-life STV elections, some votes that are set to be transferred cannot be and the number of votes still in play at the end is lower than the number of votes cast and counted in the 1st round. As well, the Droop quota is usually used in real-life STV elections. With the Droop quota in effect and five seats, it would have taken 17 percent to be elected with quota, not 20 percent as under the Hare quota.

In this case, as in all STV elections, about 80 percent or more of the votes were used to actually elect someone. A majority of the members elected in the district represent the sentiments of a majority of the voters.

This result differs from the one that would have occurred if the voting system used had been non-PR, such as single non-transferable vote (SNTV), first-past-the-post (FPTP) in five districts, first-past-the-post at-large (as used to elect members of the U.S. electoral college), or a single-winner majoritarian system in five districts

Under majority block voting, if voters voted along party lines, every Party A candidate would have received a vote from 48 percent of the voters, and some even up to 55% if voters of Candidate I also voted for some Party A candidates with their 4 other votes. At the same time, Party B's candidates could only get up to 52% of the votes with the same tactics. If the voters are partisan enough, the likely outcome is that party A would take all the seats although Party A took less than half the votes (minority representation) and all other votes are wasted.

In single-winner systems, whether First past the post or majoritarian, the outcome is uncertain. It likely would be that Party A with 48 percent of the votes might achieve a clean sweep of all five seats or easily Party A might take four of the five with Party B taking just one. (The first case would have been achieved by Party B votes being cracked by the district boundaries; the second case would have been achieved by Party B voters being mostly packed into just one district, leaving Party A with easy victories in the other four districts.) On the other hand if districts were drawn in different fashion, Party A and Party B might have divided the seats in a three to two ratio. Even under certain circumstances, the Independent candidate might take a seat if their supporters are sufficiently concentrated in one district.

STV election results are roughly proportional (as much as the number of seats allows) and take into account more than the first preferences of voters. Under STV (as seen in the example above), when it comes to secondary preferences, some voters who like a candidate from a certain party best might prefer an independent (or even a rival party candidate) before other candidates of their first choice's party. This means that even if it seems that some faction (based on first preferences) is over-represented or under-represented in the outcome, the outcome actually closely adheres to a combination of the first preferences of many voters and secondary preferences of most of the other voters. Under STV, about 80 percent of voters see their vote actually used to elect someone they prefer (and even more than that portion see someone they prefer elected even if their vote itself was not used to elect anyone), while under FPTP, often less than half of the votes are used to elect anyone and only the largest group in each district is represented.

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 goals of not only the representation of a majority of voters through the election of multiple officials but also of proportional representation of all the substantial voting blocks in the district.

Single non-transferable vote (SNTV) produces much the same representation as STV, without the work and complication of preferential ballots and vote transfers. Single voting in a multiple-member district produces mixed roughly proportional representation, which STV's vote transfers sometimes does not alter. (An example was the . The winners were the same under STV as would have been elected under SNTV.)

STV uses preferential votes cast in multi-seat districts, but some use the term "preferential voting" when they are talking only about instant-runoff voting. "Preferential voting" can also refer to a broader category, ranked voting systems. In the United States, STV is sometimes also called preferential voting, choice voting, preference voting or multi-winner ranked choice voting.

STV used for multi-winner elections is sometimes called "proportional representation through the single transferable vote", or PR-STV or STV-PR (in Scotland). "STV" usually refers to the multi-winner version, as it does in this article.

In STV, each voter casts just one vote although multiple seats are to be filled in the district. Voters mark first preference and can provide alternate preferences to be used if needed. 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.

They may indicate their preferences by ranking the candidates in order of preference. They would use ordinal numbers (1. 2. 3. etc.) to show this.

An alternative way to mark preferences for candidates is to use columns for the voters' preference with the name of each candidate appearing in each column. The first column is used to indicate 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.

Some balloting systems allow ticket voting, where a voter simply indicates preference for a party slate, sometimes even ranking party slates, instead of marking preferences for individual candidates.

Some systems declare a ballot spoiled if it is not marked with a minimum amount of preferences. In the ballot for the ACT election, voters are told they must mark at least five preferences if the ballot is to be counted. Sometimes a voter is allowed to mark just their first preference (plump) and not mark any more. Rules vary. Even where alternate preferences are marked, in some cases they may not be consulted at all, such as if the first preference candidate is elected at the end of the count to fill the last seat.

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.

In some implementations, a quota is simply set by law – any candidate receiving a given number of votes is declared elected. This system was used in New York City from 1937 to 1947. Under such a system, the number of representatives elected varies from election to election with voter turnout.

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 the ballots 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.

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:

There are variations in conducting transfers (see § Quota and vote transfers).

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.

Droop being relatively low means that the largest party, if it has majority of votes, is likely to take the majority of the seats in a district.

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.

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 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 § Filling seats under STV for details.

A quota set lower than Droop is sometimes workable. If fractional votes are used in an STV method, the Droop quota may be modified so that the fraction is not rounded down.

In any case, in most STV elections the appearance of non-transferable votes means that the quota could be lowered significantly during the counting of the vote with no danger of having too many elected.

In STV, vote transfers are of two types – 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.

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.

The transfer of surplus votes of an elected candidate may be very simply done or may be done more or less 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 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 is needed in order 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 a simpler method or more involved methods.

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

This can produce fractional votes, which are handled differently under different counting methods.

As well, not considering later preferences when transferring votes may influence later transfers and are thought of as being random, so instead some places use systems that break down the elected candidate's votes into many separate piles looking at all the permutation of preferences marked on the votes.

In some STV variants, such as those used in the Republic of Ireland (except Senate elections), Malta, and elsewhere, merely the next preference is examined. Votes are transferred as whole votes. Any randomness may arise from the later preferences, if any, that may have to be used later. But choosing the votes at random from the pile means that each transfer should be mixed and likely closely resembles the composition of the entire pile.

As a result of a parliamentary commission investigating the 2013 election, from 2016 the system has been considerably reformed (see 2016 Australian federal election), with group voting tickets (GVTs) abolished and voters no longer required to fill all boxes.

Less well known is STV use at the municipal level in western Canada. Calgary and Winnipeg used STV for more than 50 years before city elections were changed to use the first-past-the-post system. Nineteen other municipalities, including the capital cities of the other western provinces, also used STV For elections in about 100 elections during the 1918 to 1931 period.

The table below lists countries that use STV to fill a nationally elected legislative body by direct elections.

The degree of proportionality of STV election results depends directly on the district magnitude (i.e. the number of seats in each district). While Ireland originally had a median district magnitude of five (ranging from three to nine) in 1923, successive governments lowered this. Systematically lowering the number of representatives from a given district directly benefits larger parties at the expense of smaller ones.

Supposing that the Droop quota is used: in a nine-seat district, the quota or threshold is 10% (plus one vote); in a three-seat district, it would be 25% (plus one vote). This electoral threshold is significantly higher than for most party-list PRs.

STV provides proportionality by transferring votes to minimize waste, and therefore also minimizes the number of unrepresented or disenfranchised voters.

The algorithm is complicated. In large elections with many candidates, a computer may be required. (This is because after several rounds of counting, there may be many different categories of previously transferred votes, each with a different permutation of early preferences and thus each with a different carried-forward weighting, all of which have to be kept track of.)

STV differs from other proportional representation systems in that candidates of one party can be elected on transfers from voters for other parties. Hence, STV may reduce the role of political parties in the electoral process and corresponding partisanship in the resulting government. A district only needs to have four members to be proportional for the major parties, but may under-represent smaller parties, even though they may well be more likely to be elected under STV than under first-past-the-post.

As STV is a multi-member system, filling vacancies between elections can be problematic, and a variety of methods have been devised:

The outcome of voting under STV is proportional within a single election to the collective preference of voters, assuming voters have ranked their real preferences and vote along strict party lines (assuming parties and no individual independents participate in the election). However, due to other voting mechanisms usually used in conjunction with STV, such as a district or constituency system, an election using STV may not guarantee proportionality across all districts put together.

A number of methods of tactical or strategic voting exist that can be used in STV elections, but much less so than with first-past-the-post elections. (In STV elections, most constituencies will be marginal, at least with regard to the allocation of a final seat.)

STV systems vary, both in ballot design and in whether or not voters are obliged to provide a full list of preferences.

Allowing voters to rank only as many candidates as they wish grants them greater freedom, but can also lead to some voters ranking so few candidates that their vote eventually becomes "exhausted"– that is, at a certain point during the count, it can no longer be transferred and therefore loses an opportunity to influence the result. However the number found to be non-transferable under STV is less than are ignored or wasted under the first-past-the-post system. Some votes found to be non-transferable are that way because the choices marked have already been elected, so the voter may be pleased with the overall election result even though their first preference was not elected and their vote itself was not used to elect anyone. Even if a voter marks many alternate preferences, the vote will still be found to be non-transferable, if at any point the vote needs to be transferred and all the preferences listed next have already been eliminated or elected. But the number of non-transferable votes is fewer than the number of ignored votes under first-past-the-post and the number of effective votes, votes actually used to elect someone, is higher than under all but the most landslide first-past-the-post election contests.

The STV method can be confusing, and may cause some people to vote incorrectly with respect to their actual preferences.

STV ballots can also be long; having multiple pages increases the chances of people not marking multiple preferences and thus missing later opportunities to have their vote transferred. However, after a vote has been transferred a couple times and it now is at the end of the count and just three candidates remain in the running for the last seat, the voter may have little interest in the choice. None of them were the voter's first choice, nor their second or third preference. And perhaps the voter has already seen one or two of their earlier choices already elected. Many votes up for transfer are found to be non-transferable in the last vote transfers. Some at the end are elected with partial quotas, due to the number of non-transferable votes. But in STV elections a majority of votes are used to elect the members who are elected.

The relative performance of political parties in STV systems is sometimes analysed in a different fashion from that used in other electoral schemes. For example, seeing which candidates are declared elected on first-preference votes alone in the 2012 Scottish local elections, where 1223 members were elected, can be shown as follows:

The transfers of votes under STV mean that candidates who did well on first-preference votes in the first count (but not well enough to be immediately declared elected) may not be elected in the end, and those who did poorly on the first count may be elected in the end. This is due to transfers made according to second and later preferences. This can also be analysed, again using the 1223 members elected in the Scottish local elections. Some of the leading candidates in the first count were not elected but, comparing the number to the total number of members elected in these elections, the successful candidates were mostly set in the first count (through the simple mechanics of single voting in multi-member districts), before any vote transfers are done. About ten percent or less of the front runners in the first count were not elected in the end.

Sixty-eight of the elected members, of the overall 1,223 successful candidates, were not already in winning position in the first count, thus showing that vote transfers merely put a polish on the first-count ranking of candidates established through single voting in multi-seat district.