# Single transferable vote

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The single transferable vote (STV) is a preferential voting system designed to minimize wasted votes in multi-candidate elections while ensuring that votes are explicitly for candidates rather than party lists. It works by assigning candidates votes based on the number of voters who ranked them 1st, electing candidates who reach a certain threshold of votes ("quota") and spending those votes to ensure as-of-yet unrepresented voters can get someone they like, and otherwise eliminating the candidate with the fewest 1st choices and then treating the uneliminated candidate their voters ranked next-highest as their "new" 1st choice.

When promoted as a proportional representation method in multi-party multi-seat elections, it is generally known as Proportional Representation through the Single Transferable Vote or PR-STV. When a similar method is applied to single-seat elections it is sometimes called instant-runoff voting or the alternative vote, and has different proportionality implications for a similar ballot.

## Voting

Each voter ranks all candidates in order of preference. For example:

1. Andrea
2. Carter
4. Delilah

### Setting the Quota

When all the votes have been cast, a winning quota is set. The most common formula for the quota is the Droop Quota which is most often given as:

floor(votes / (seats + 1)) + 1

Other quotas used include the Hare Quota:

and the Imperiali Quota:

For those keeping track, the size of the quota is then generally Hare > Droop > Imperiali.

Process A: Top-preference votes are tallied. If one or more candidates have received at least as many votes as the quota, they are declared elected. After a candidate is elected, they may not receive any more votes (though see below for a modernisation).

The excess votes for the winning candidate are reallocated to the next-highest ranked candidates on the ballots for the elected candidate. There are different methods for determining how to reallocate the votes. Some versions use random selection, others count each ballot fractionally.

Process A is repeated until there are no more candidates who have reached the quota.

Process B: The candidate with the least support is eliminated, and their votes are reallocated to the next-highest ranked candidates on the eliminated ballots. After a candidate is eliminated, they may not receive any more votes.

After each iteration of Process B is completed, Process A starts again, until all candidates have been elected or eliminated.

### An example

2 seats to be filled, four candidates: Andrea, Brad, Carter, and Delilah.

5 voters rank the candidates:

1. Andrea
3. Carter
4. Delilah

17 voters rank the candidates:

1. Andrea
2. Carter
4. Delilah

8 voters rank the candidates:

1. Delilah

The threshold is: floor(30 / (2 + 1)) + 1 = 11.

In the first round, Andrea receives 22 votes and Delilah 8. Andrea is elected with 11 excess votes. Her 11 excess votes are reallocated to their second preferences (which votes are chosen may be decided by random selection). For example, 8 of the reallocated votes are for Carter, 3 for Brad. Note: this is not a realistic example - elections with a small number of votes often have special rules - for example, Irish Senate elections are conducted using thousands of votes.

As none of the candidates have reached their threshold, Brad, the candidate with the fewest votes, is eliminated. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 11 votes and he is elected.

## Is STV a proportional voting system?

STV is not a proportional system in the strict sense. STV does not guarantee that a party will get the same percentage of seats as it gets as a percentage of votes. In fact the notion of a vote "for a party" is less meaningful for STV because votes are not necessarily for a single party. A vote can list candidates from an assortment of political parties, in any order. The candidates that are elected reflect the combined preferences of all votes cast.

Another complication with proportionality under STV is the constituency system, where a set of candidates is elected in each electoral district. There is no explicit process in STV for balancing the votes between constituencies, so the overall electoral result is merely the sum of the constituency results.

Within a constituency, however, STV can be said to be proportional for whatever characteristics the voters valued. For example, if 60% of voters put all the female candidates first, and 40% put all the male candidates first, 60% of the winners would be female and 40% would be male. (Assuming there are sufficient candidates of each gender to make up the numbers.)

STV provides this proportionality simply by wasting as few votes as possible. A vote is a "Wasted vote" if it does not elect anyone; it is partially wasted if it elects someone who gets more votes than is necessary to be elected. STV transfers votes that would otherwise be wasted, and it only transfers such votes.

The degree of proportionality nationwide is strongly related to the number of seats to be filled in each constituency. In a three-seat constituency, using the Droop quota, about a quarter of the vote is "wasted". These votes may be for minor candidates that were not eliminated, or elected candidates' surplus votes that did not get redistributed. In a nine-seat constituency, only a tenth of the vote is wasted, and a party needs only 10% of the vote in a constituency to win a seat. Consequently, the best proportionality is achieved when there are a large number of representatives per constituency.

The proportionality of STV can be controversial, especially in close elections such as the 1981 election in Malta. In this election the Maltese Labour Party won a majority of seats despite the Nationalist Party winning a majority of first preference votes. This caused a constitutional crisis, leading to provision for the possibility of bonus seats. These bonus seats were used in 1987 and again in 1996. Similarly, the Northern Ireland elections in 1998 led to the Ulster Unionists winning more seats than the Social Democratic and Labour Party, despite winning a smaller share of the vote.

Advocates of STV argue that the apparent disproportionality in STV is indicative of poor support for the party's candidates in second and third preferences. They argue that the STV result is actually a more accurate estimate of the party's support than a simple tally of first-preference votes.

Within each constituency, STV passes the Droop proportionality criterion when using Hagenbach-Bischoff quotas (when doing so, it is suggested that a candidate win only if they exceed quota), because when all but k candidates of a solid coalition's supported candidates have been eliminated, one of the remaining candidates is now the 1st choice of over a quota of voters, and is thus elected, with their surplus votes flowing towards some of the other k candidates, and this repeating, until all k candidates are elected. This guarantees a majority will win at least half of the seats in each constituency.

## Potential for Tactical Voting

The single transferable vote eliminates much of the reason for tactical voting. Voters are "safe" voting for a candidate they fear won't be elected, because their votes will be reallocated in Process B. They are "safe" voting for a candidate they believe will receive overwhelming support, because their votes will get reallocated in Process A.

However, in older STV systems there is a loophole: candidates who have already been elected do not receive any more votes, so there is incentive to avoid voting for your top-ranked candidate until after they have already been elected. For example, a voter might make a tactical decision to rank their top-place candidate beneath a candidate they know will lose (perhaps a fictional candidate). If the voter's true top-place candidate has not been elected by the time their fake top candidate loses, the voter's full vote will count for their true top-place candidate. Otherwise, the voter will have avoided having had their ballot in the lottery to be "Wasted Votes" on their top-ranked candidate, and will continue on to lower-ranked candidates.

Note that in more modern STV systems, this loophole has been fixed. A vote receives the same fractional weighting regardless of when it arrives at the successful candidate. This modernisation has not been adopted in all STV systems.

There are also tactical considerations for parties standing more than one candidate in the election. Standing too few may result in all the candidates being elected in the early stages, and votes being transferred to candidates of other parties. Standing too many candidates might result in first-preference votes being spread amongst them, and several being eliminated before any are elected and their second-preference votes distributed, if voters do not stick tightly to their preferred party's candidates; however, if voters vote for all candidates from a particular party before any other candidates and before stopping expressing preferences, then too many candidates is not an issue - in Malta where voters tend to strictly express party preference, parties frequently stand more candidates than there are seats to be elected.

Vote management is a potential strategy in STV that involves bullet voting in a way that approximates D'Hondt to maximize a party's seat share. 5-winner example:

51 (Party A candidates)

49 (Party B candidates)

10 (Party C candidates)

10 (Party D candidates)

Using an HB quota (120/6= 20 votes), 2 Party A candidates and 2 Party B candidates are elected, leaving Party A with 11 votes and B with 9. This allows Party C or D to win the final seat. However, if the Party A votes had instead been:

17 A1

17 A2

17 A3

2 Party B candidates win as before, but now every party except the A candidates has fewer than 17 votes, so all other candidates are eliminated one by one such that A1-3 win the final 3 seats. Schulze STV is designed to give Party A 3 seats in this example even without vote management.

## In practice

Places that use STV for governmental elections include:

• Australia, for the Senate [1] and for one or other of the state houses.
• Ireland, for all elections [2]
• Malta, for all elections [3]
• New Zealand [4], where STV is being used for the first time for district health board and some local authority elections in October 2004
• Northern Ireland, for local, Assembly and European elections
• The United States, where the only official governing bodies that use STV to elect representatives are the City Council and School Committee of Cambridge, Massachusetts.

STV enjoyed some popularity in the United States in the first half of the 20th Century. The community school boards of the City of New York [5] used STV until they were abolished in 2002.

The method used for electing the Legislative Assemblies of Tasmania and the elections in the province of Alberta, Canada from 1926 to 1955.

British Columbia will decide in 2005 by referendum whether to adopt STV to replace its current First Past the Post electoral system, after a recommendation of STV [6] by the Citizens' Assembly on Electoral Reform.

Some non-governmental organisations also use STV. For instance, all National Union of Students of the United Kingdom elections and those of their constituent members are under the system.

## Historical assessments

An early proponent of STV was John Stuart Mill, who praised it in "On Representation." In the "English Constitution" Walter Bagehot praised the Hare system for allowing everyone, even ideological minorities, to elect an MP, but said that the Hare would create more problems than it solved. "[the Hare system] is inconsistent with the extrinsic independence as well as the inherent moderation of a Parliament - two of the conditions we have seen, are essential to the bare possibility of parliamentary government."

## Variations

### Ways of dealing with equal rankings

• Disallowing them, requiring full rankings
• Giving one vote to one of the top-ranked candidates, and giving no vote to the other top-ranked candidates (likely best done by choosing randomly which top-ranked candidate gets the vote)
• ER-IRV (fractional): Counting a ballot with N top-ranked candidates as 1/N of a vote for each candidate.
• ER-IRV (whole votes) or Approval-IRV: Giving one vote to each equally-top-ranked candidate. Can optionally be combined with a suggestion that ballots that equally rank candidates shouldn't be able to prevent the elimination of those candidates.[1]

The nature of IRV is such that many of its derived properties depend on each voter exerting influence such that their support for all candidates adds up to at most "one vote" (see Category:FPTP-based voting methods). Thus, fractional equal-ranking preserves most of IRV's criterion compliances and properties, but not Approval-IRV. In addition, fractional equal-ranking creates less ambiguity around how to configure the IRV rules, whereas with Approval-IRV, different variations of the rules which would give the same results with other forms of IRV can give different results:

Note that in the single-winner case, giving a vote to each equally-top ranked candidate can lead to different results when using either the "if in any round any candidate gets a quota (if using a Droop quota, a majority, in the single-winner case) they win" rule or the "all but ((number of winners) + 1) candidates must be eliminated, with the (number of winners) candidate(s) with the most votes then winning" rule (which can also be thought of as "all but (number of winners) candidates must be eliminated, with the remaining candidate(s) winning"). Single-winner example:

45 A=C>B

35 B>A>C

20 C>B>A

If the first rule is used (modified to also say "if multiple candidates have Droop quotas in the same round, the candidate(s) with the largest Droop quotas win"), then C wins with 65 votes to start off with, whereas under the second rule, B is eliminated, and then A wins. The second rule can actually create greater possibility for pushover strategy, since it could have been the case the A=C>B voters' honest preferences were A>B>C, and if they had voted their honest preference, C would've been eliminated and then B would've won, a worse result from their point of view. Also, this can lead to different results when using either the "all candidates who reach quota are elected" rule or the "the candidate who most exceeds the quota is elected, then spend their ballots, and repeat" rule. 3-winner example with Droop quotas:

34 A=B=C

33 D

33 E

A, B, and C win under the former rule, while A, D, and E win under the latter rule.

### Methods of transferring excess votes

If the quota is 100 votes and a candidate wins with 250 votes:

• Random transfer: 100 ballots are randomly chosen from the pool of 250 and "spent", with the other 150 being transferred to their next-highest-uneliminated preferred candidate.
• Fractional transfer: 100/250=40% of all 250 ballots is spent, with all 250 ballots now supporting their next-highest-uneliminated preferred candidate, but with only 60% ballot weight/power. There are several possible variations of Surplus Handling

### Ways of choosing a candidate to eliminate

• Standard STV: Eliminate the candidate with the fewest top-choice votes.
• BTR-STV: Eliminate the pairwise loser of the bottom two candidates (meets Condorcet criterion in single-winner elections).
• Benham: If the candidate with the fewest top-choice votes beats every other remaining candidate pairwise, eliminate the candidate with the next fewest top-choice votes. Meets the Condorcet criterion in single-winner elections.
• STV-CLE: Eliminate the loser of a Condorcet ranking or a Condorcet method's ranking.

### Deciding the election of the final seat

The following pseudocode template can be used to run STV with any single-winner method used to elect the final seat:

So long as there are still seats to be filled, and the number of seats remaining to be filled is more than the number of unelected candidates (eliminated or uneliminated) left:

1. Count the number of 1st choices for each candidate.
2. If any candidate has as many or more votes than the quota, elect them, and spend as much of a quota as possible of their votes. Distribute any surpluses (votes over the quota) that the just-elected candidates had to their voters' next preferences. Repeat until no more candidates can be elected. If there are as many uneliminated candidates left as seats to be filled, elect all of them except the one with the fewest votes.

If only one seat is unfilled, go to step 3. Otherwise, go to step 4.

3. "Un-eliminate" all previously eliminated candidates*, and run any single-winner method, treating ballots with partial weight as partial ballots i.e. a ballot A>B>C with 60% weight remaining in the STV count is treated as 0.6 A>B>C ballots for the single-winner method's count. Elect the winner of the single-winner method.

4. Eliminate the candidate with the fewest 1st choices and redistribute their voters' votes to their next preferences. Go to step 2.

• Conceptually speaking, the candidates who are eliminated during the STV portion of the count could be said to be eliminated from contention for the first N - 1 seats, but still remain in contention for the final seat; though technically, it's possible for the final seat to be filled during the STV count itself (i.e. in a 2-seat election, 2 candidates might each have a quota of 1st choices and automatically win).[2]

Note that with this scheme, after all but one seat is filled, surplus distribution is only necessary when there is enough of a surplus to potentially help one of the uneliminated candidates get a quota.

If the single-winner method passes the majority criterion, then this modification makes STV with Droop quotas become the single-winner method in the single-winner case, since a candidate with a Droop quota of 1st choices is the majority's 1st choice in the single-winner case. If STV with Hare Quotas is used instead, then this modification can be used with any single-winner method that passes unanimity and reduce to that single-winner method in the single-winner case.

5-winner example:

45 L1 > L2 > L3 > C > R2 > R1

10 C > L3 > R2 > L2 > L1 > R1

45 R1 > R2 > R3 > C > L2 > L1

The first 4 winners are (L1, L2, R1, R2). If the Droop quota (17 votes) is used, the remaining ballots after spending the ballots supporting the first 4 winners will be (ignoring elected candidates):

11 L3 > C

10 C > L3

11 R3 > C

Standard STV eliminates C and transfers their 10 votes to L3, making L3 the final winner. However, C is the Condorcet, Bucklin, and Borda winner, and would win if any of those methods were used to elect the final seat. Note that electing L3 for the final seat would mean that a 45% minority of voters would have 60% of the seats, whereas with C being elected for the final seat, a majority of voters are guaranteed to have a majority of the seats. For STV to still pass Droop proportionality with this modification, the single-winner method must pass at least mutual majority (since that is Droop proportionality in the single-winner case). 2-winner example:

51 E

19 C>A>B>D

17 B>C>A>D

16 A>B>C>D

17 D>C>A>B

16 D>B>C>A

15 D>A>B>C

E has a Droop quota (51 votes) so they win, and their 51 supporters' ballots are spent. Then we run a majority-passing single-winner method, say, Minimax on the remaining 100 ballots, and D wins despite there being a Droop solid coalition and thus a mutual majority of 52 voters for (A, B, C). Only a mutual majority-passing single-winner method would guarantee one of A, B, and C wins. This modification to STV addresses to some extent one of the criticisms made by STV advocates about single-winner methods other than IRV:

Furthermore, approval, score, and Condorcet were all designed to be used in single-winner elections only. Ranked choice voting works well for both single-winner and multi-winner elections. For elections that involve a mixture of single-winner and multi-winner races, we strongly prefer the simplicity of using a uniform voting method across the board.[3]

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## Notes

STV is a largest remainder method in the party list case. This is because it gives each party as many seats as they have quotas, and when nobody can reach the quota anymore, it eliminates candidates until there are only as many candidates left as seats that are still unfilled, which de facto means that the parties with the most votes remaining (i.e. largest remainders) will win, since they'll be last to be eliminated.

STV and IRV can be visualized using Sankey or flow diagrams. If, in a given round, a candidate reaches the quota (a majority in IRV), they win, and if more seats are to be filled, then a quota of their votes are spent. This could be visualized by arranging the candidates from most votes to fewest in each round, and then showing a threshold for how many votes the candidate with the most votes in that round needs to win, or even further, showing the quota needed for the top (number of seats to be filled) candidates to win. Here is one example: [4]

STV passes PSC regardless of the method used to decide which candidate is to be eliminated next, because for a solid coalition comprising k quotas, once all but k of the candidates in the coalition are eliminated, at least one of the k remaining candidates will have a quota of 1st choices and win, with their surplus transferring if necessary such that another of the k candidates wins, etc. It passes the Droop proportionality criterion when the Droop quota is used, making it one of the few commonly discussed PR methods that guarantees that a majority will always win at least half of the seats.

STV has several variations that can be discussed. For example, Meek and Warren STV are variants of STV that attempt to make the process fairer, but at the cost of needing to be computerized to compute the result.

There is some discussion regarding how to make STV results more transparent, while limiting the ability of vote-riggers to identify specific voters by having access to all of the preference data. [5]