Single Transferable Vote (STV)

Summary: For electing a group of candidates, the single transferable vote is a great method because it provides proportional representation. E.g., if a group has 40% support, it should be able elect 40% of the winners. The voters rank the candidates, and the votes are transferred to find a set of winners that best match the voters' preferences.

The single transferable vote is a type of ranked-choice voting that is used for electing a group of candidates (e.g., a committee or council). When used to elect one candidate, it is basically the same as instant runoff voting. On the right is an example ranked ballot for an online election.

The single transferable vote is similar to instant runoff voting in that votes are transferred from losing candidates to other choices on the ballot. STV also has a second kind of vote transfer. A candidate in an STV election can have too many votes, called surplus votes, and these surplus votes can also be transferred to other candidates.

One of the main principles of the single transferable vote is to obtain proportional representation. Proportional representation means, in a rough sense, that the demographics of the elected body should mirror the demographics of the electorate. Transferring surplus votes is a key part of ensuring proportional representation.

The easiest way to explain is with an example. Suppose a highly partisan town is electing a city council of 5 councillors using STV. The town's voters are 80% Republican and 20% Democrat. If we want proportional representation, then 4 of the 5 city councillors should be Republican.

The city has 100 voters, the Republicans have 5 candidates, and the Democrats have 3 candidates. For the Republicans, one candidate is hugely popular and gets most of the Republican first choices. Accordingly, when votes are transferred to their first choices, we have the following distribution.

Round R1 R2 R3 R4 R5 D1 D2 D3 1 75 2 1 1 1 8 7 5

If we stop here then the Republicans win 2 seats and the Democrats win 3 seats even though 80% of the city is Republican. We could start eliminating candidates with the fewest votes, but this doesn't help the Republicans, because we would just be eliminating Republican candidates.

The solution is to transfer the surplus votes of candidate R1. To figure out how many surplus votes that R1 has, we establish a "winning threshold" using this formula and then dropping any fraction:

threshold = number of votes number of seats + 1 + 1

For this election, the winning threshold is 100/(5+1) + 1 or 17. Accordingly, candidate R1 has 75 - 17 = 58 surplus votes. Here is what the next round of counting might look like after we transfer R1's surplus votes:

Round R1 R2 R3 R4 R5 D1 D2 D3 1 75 2 1 1 1 8 7 5 2 17 27 16 13 7 8 7 5

Now, this is starting to look more reasonable! At this point, it looks like the Republicans will take 4 of the 5 seats as expected.

Note that candidate R2 now has surplus votes so those surplus votes would be transferred next. This process continues until 5 candidates reach the winning threshold or until only 6 candidates remain. We'll skip the intervening rounds transferring surplus votes and eliminating candidates, but the final result could look something like this:

Round R1 R2 R3 R4 R5 D1 D2 D3 1 75 2 1 1 1 8 7 5 2 17 27 16 13 7 8 7 5 ... 5 17 17 17 17 12 17 3 0

In the end, we are electing 4 Republicans and 1 Democrat so we have proportional representation. While this was an extreme example to illustrate how STV works, STV provides proportional representation in less extreme situations as well.

More generally, counting votes with STV proceeds as follows:

Count the first place votes. Transfer votes from one candidate to other candidates:

If a candidate has surplus votes (votes in excess of the winning threshold), then transfer surplus votes to their next choices.

Otherwise, eliminate the last place candidate and transfer those votes to their next choices.

If not all seats have been filled, then go to step 2.

Each of the STV methods below specify additional details (or modify) these three steps.

STV is used for government elections in Cambridge, Massachusetts; Minneaoplis, Minnesota; Scottland; Australia; Ireland; N. Ireland; and Malta.

OpaVote provides several versions of STV, such as Scottish STV, Meek STV, ERS97 STV, and Minneapolis STV. You can see an example of STV results generated by OpaVote.

Scottish STV

The Scottish STV rules are recommended for most organizations because the rules are well defined and provide a straightforward implementation of STV that is easier to understand. Scotland enacted STV in 2007 and had its first election that year. Our blog post provides a plain English explanation of the Scottish STV rules.

Scottish STV has the following features:

Single for Candidate Elimination

Static and Whole Threshold

No delayed transfer of surplus votes

Precision of 5

Backward for Tie Breaking

Meek STV

Meek STV is recommended for organizations whose members are comfortable with a more complicated counting method. Meek STV provides more accurate proportional representation than other STV methods, but takes more effort to understand.

One advantage of Meek STV is that when a candidate is eliminated from the election, the votes are counted as if the candidate was never in the election at all and the order of elimination cannot effect the outcome. With other STV methods, the order of elimination can effect the outcome.

Another advantage of Meek STV is that surplus votes are transferred in a better way. With other STV methods, surplus votes are not ever transferred to a candidate who has already won. With Meek STV, surplus votes are always transferred to the next candidate on the ballot.

For additional details about Meek STV, see our blog post explaining Meek STV or first issue of Voting Matters.

ERS97 STV

The Electoral Reform Society of the United Kingdom has been providing STV rules since at least as early as 1955 and its latest rules from 1997 are commonly referred to as the ERS97 STV rules. These rules are widely used in the UK. The ERS97 rules are the most complicated of all the STV rules provided by OpaVote. For this reason, we recommend other STV rules.

Minneapolis STV

Minneapolis enacted STV in 2006 and had its first STV election in 2009. The Minneapolis rules are very similar to the Scottish rules.

Minneapolis STV has the following features:

Losers for Candidate Elimination

Static and Whole Threshold

Delayed transfer of surplus votes

Precision of 4

Random Tie Breaking

N. Ireland STV

The N. Ireland STV rules are similar to the ERS97 rules, but significantly simpler.

Warren STV

Warren STV is very similar to Meek STV. To learn more about the differences, see the first issue of Voting Matters.

Cambridge STV

The City of Cambridge, Massachusetts has used Cambridge STV rules to elect its city council and school committee since 1941. Note that the statute allows Cambridge to use any method for transfering surplus votes that was in use in 1938, and Cambridge has chosen to use the Cincinnati method.

Since candidates with fewer than 50 votes are eliminated, this method should not be used with a small number of ballots.

The City of Cambridge describes the Cincinnati method as follows:

The ballots of the candidate who has a surplus are numbered sequentially in the order in which they have been counted (that is, in the sequence dictated by the random draw of precincts) and then every nth ballot is drawn and transferred to a continuing candidate until the original candidate is credited with ballots equaling no more than quota. n is nearest whole number computed by the formula

n = Candidate's Total Ballots

Surplus Ballots.

A ballot selected by this method that does not show a preference for a continuing candidate is skipped and remains with the original candidate. If not enough ballots are removed when ballots n, 2n, 3n, .... have been transferred, the sequence starts again with n+1, 2n+1, 3n+1, ....

Fractional Transfer STV

The fractional transfer STV method is a generalization of the Scottish STV rules, and you can use the options to customize your counting rules (but note that these options are available only for counts and not for elections and polls).

Random Transfer STV

Except for Cambridge STV, all of the STV counting rules above use fractional votes for transferring surplus votes. This is a general method that allows you transfer surplus votes as whole votes instead of fractional votes. This method is not recommended for actual elections, but you may find it interesting to compare with other methods.

Note that the transfers of votes are not actually random, but the outcome of an election can depend on the order of the ballots. If you shuffle the ballots and recount, you could obtain a different winner. All of the methods above (except Cambridge STV) will always produce the same winners after shuffling ballots.