Counting single transferable votes

Counting Single Transferable Votes in Single Transferable Vote elections has many variations.

Voting

If a class of children were choosing representatives, say, they could line-up behind the candidate of their choice. Since they would all know that each candidate only needs a certain number of classmates to vote for them to be elected, those arriving last in line for a candidate who already has enough votes would choose to not waste their vote and would instead move to another line to help someone else to win. Likewise, those children whose candidate obviously could not win, would move to another line, and so on, until all the representatives are chosen.

When using an STV ballot, these preferences are set out in advance, as instructions to the counters.

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

1. Andrea
2. Carter
3. Brad
4. Delilah

Setting the quota

Choice of quota

The quota (sometimes called the threshold) is the number of votes a candidate must receive to be elected. The Droop quota is preferred because it is the smallest number that ensures, if as many candidates as there are seats to be filled each have a quota of votes, no other candidate can have a quota. A candidate's surplus votes are transferred to other candidates according to the next available preference. In Meek's method the quota must be recalculated throughout the count.

With the Hare quota even if each voter expresses a preference for every candidate, at least one candidate is likely to be elected with less than a full quota. If each voter expresses a full list of preferences, the Droop quota guarantees that every candidate elected will meet the quota rather than be elected by being the last remaining candidate after lower candidates are eliminated.

Droop quota

The most common formula for the quota used now is the Droop Quota which is most often given as:

$\left({{\rm {votes}} \over {\rm {seats}}+1}\right)+1$.

Unlike the Hare Quota, this does not require that all preferences must reach a final home. It is only necessary that enough votes be allocated to ensure that no other candidate still in contention could win. This leaves nearly a quota's worth of votes unallocated, but it is held that this quota simplifies voting, and that counting these votes would not alter the eventual outcome.

Hare quota

When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:

${\rm {votes}} \over {\rm {seats}}$

This has thus become known as the Hare Quota. It would require that all votes cast be divided equally between the eventually successful candidates. The only differences, thus, between the votes received for each candidate would be based on the distribution of voters between constituencies (Hare's original proposal was for a single national constituency) and the number of non-continuing votes, i.e. people who did not express a preference for all candidates, meaning that some candidates would be elected with less than a quota as the last remaining.

Counting the votes

Process A: Top-preference votes are tallied. If one or more candidates receive at least as many votes as the quota, they are declared elected. After a candidate is elected, she 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 excess votes. (See below)

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 his votes are reallocated to the next-highest ranked candidates on the eliminated ballots. After a candidate is eliminated, he 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.

Surplus re-allocation

Hare's method

With the school children (above) all the votes that arrived after the candidate reached a quota of votes were simply redirected to their next preferences. The same can be done with ballot papers, on the rough assumption that the votes in the surplus are a representative sample of all the ballots cast. The possibility remains, however, that they will not represent a properly mixed sample.

Cincinnati method

The city of Cambridge, Massachusetts, uses a method called the Cincinnati method to increase the likelihood of a representative spread of transfers. The votes from each polling district are counted and merged together into a randomised sequence. Then, taking the formula

$\left({\frac {\mathrm {Candidate's\ Total\ Ballot} }{\mathrm {Candidate's\ Surplus} }}\right)=n$

every nth ballot is removed from the candidates pile and transferred. If no continuing preference is stated on the paper, then the next available ballot is chosen. If the candidate's votes are not reduced to the quota after removing every nth ballot, every (n+1)th ballot is removed.

Clarke method

In Australia, A.I. Clarke (see below) devised the method whereby all a candidate's next preferences are counted, and their surplus is transferred according to the various ratios of such preferences found. e.g. if a candidate has 100 surplus votes, and 25% of all their votes have the same next preference, 25 will be allocated accordingly.

Senatorial rules

Another method is known variously as the Senatorial rules - after its use for Irish Senate elections, or the Gregory method - after its inventor in 1880 J.B. Gregory of Melbourne. This continues Clarke's method. Instead of dividing the surplus out according to the fractions of next preferences a candidate received, all of their votes are passed on to their next preference, with their new effective value found by:

$\left({\frac {\mathrm {Quota} }{\mathrm {Candidate's\ votes} }}\right)$

To avoid confusion, the decimal point is sometimes removed, and votes are given a nominal value of 100 instead of 1.

Meek method

All of the above methods assume manual counting of ballot papers. In 1969, B.L. Meek devised a method which is suitable for computer counting.

All candidates are allocated one of three statuses - Hopeful, Elected, or Excluded. Hopeful is the default. Each status has a weighting, or keep value, which is the fraction of the vote a candidate will receive for any preferences allocated to them while holding that status.

The weightings are:

Hopeful	$1$
Excluded	$0$
Elected	$1-\left({\frac {\mathrm {Quota} }{\mathrm {Candidate's\ votes} }}\right)$

Thus, if a candidate is Hopeful they retain the whole of the remaining preferences allocated to them, and subsequent preferences are worth 0.

If a candidate is Elected they retain the value of their weighting and the remainder of the value of the vote is pass along fractionally to subsequent preferences depending on their weighting, with the formula

$\left(1-{\mathrm {nthWeighting} }\right)$

being carried out at each preference.

This results in a fractional excess, which is disposed of by altering the quota, hence Meek's method is the only method to change quota mid-process. The quota is found by

$\left({{\rm {votes-excess}} \over {\rm {seats}}+1}\right)$,

a variation on the droop quota. This has the effect of also altering the weightings for each candidate.

This process continues until all the Elected candidates' vote values almost equal the quota (within a very close range, i.e. between 0.99999 and 1.00001 of a quota). [1]

An example

Suppose we conduct an STV election using the droop quota where there are two seats to be filled and four candidates: Andrea, Brad, Carter, and Delilah. Also suppose that there are 57 voters who cast their ballots with the following preference orderings:

	16 Votes	24 Votes	17 Votes
1st	Andrea	Andrea	Delilah
2nd	Brad	Carter	Andrea
3rd	Carter	Brad	Brad
4th	Delilah	Delilah	Carter

The threshold is: $\left({57 \over (2+1)}\right)+1=20$

In the first round, Andrea receives 40 votes and Delilah 17. Andrea is elected with 20 excess votes. Her 20 excess votes are reallocated to their second preferences. For example, 12 of the reallocated votes go to Carter, 8 to Brad.

As none of the remaining candidates have reached the quota, 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 20 votes and he is elected, filling the second seat.

Thus:

	Round 1	Round 2	Round 3
Andrea	40	20	20	Elected in round 1
Brad	0	8	0	Eliminated in round 2
Carter	0	12	20	Elected in round 3
Delilah	17	17	17	Defeated in round 3