The Maths of Voting

Ireland uses proportional representation (PR) for voting in elections, with each voter having a single transferable vote (STV). PR-STV is a candidate-based system. This means voters can choose to vote for as many, or as few candidates as they like, in order of their preference. The voter’s first preference vote – the candidate they give their number 1 vote to – is most important and is always counted. A voter’s second (and further preferences) may be counted if their preferred candidate is eliminated at the end of a round of counting, or is elected with a surplus. These are known as transfers. The system has an interesting history and some interesting maths !!

In Britain, the  first meetings of the Proportional Representation Society quickly attracted many leading lights of the Victorian age – including Lewis Carroll, CP Scott (editor of what is now The Guardian) and Thomas Hare (the inventor of the STV). The group quickly settled on Hare’s system as the best option.

A matter of upmost importance is the quota. To work out who gets elected, you need to work out the quota to be elected. Candidates that exceed the quota are elected, with any surplus votes (total votes minus the quota) transferred to each voter’s next choice as indicated on the ballot paper. Once any candidates who beat the quota are elected, there is another round of counting to see if any other candidates have reached the quota, now the surplus votes have been transferred. If no candidate meets the quota in a particular round of counting, the candidate with the least votes is eliminated and their votes are redistributed to their voters’ next preference. This continues until all the seats have been filled. We can all remember some marathon counts in the midlands!!

But how do you work out what the quota is? This is one of the things that has been tweaked since Hare’s original designs for STV were laid out.

Hare vs Droop – Electoral Quotas

The two main quota calculations are Thomas Hare’s original quota – which is “total votes / total seats” and Henry Droop’s quota – which is “(total votes / (total seats + 1)) + 1”.

In a constituency electing three TD’s where 960 votes have been cast, the Hare quota would be 320 and the Droop quota would be 241. While nearly all STV elections today use the Droop quota, some still advocate for Hare-STV – pointing to it typically producing slightly more proportional results and it being more favourable to smaller parties. But Hare-STV does have a number of things to consider.

There is an issue of inequality. Under Droop-STV, all elected candidates beat the quota and are thus elected on the same terms. But under Hare-STV it is practically impossible for all candidates to meet the quota. As such, the fight for the final seat is awarded to the candidate with the most remaining votes, regardless of how short they have fallen of the quota. Then there is the majority rule problem. In certain circumstances under the Hare quota, it is possible for a party to win slightly more than half of votes cast but to win less than half of seats in a constituency.

Suppose that an STV election takes place between the Offaly Party and the Laois Party, with each running two candidates for three seats and 960 voters.

  • 510 voters give their first preference to an Offaly Party candidate – 340 for the first candidate and 170 for the second candidate.
  • 450 voters choose the Laois Party, but with their voters more evenly splitting between the two candidates – 226 for the first and 224 for the second.

All voters rate both candidates from their preferred party ahead of the candidates for the opposing party, with only half of voters afterwards ranking an opposing candidate.

Under a Hare-STV election, where the quota would be set at 320 (960 voters / 3 seats), the first Offaly candidate would be instantly elected, and their 20 surplus votes transferred to the other Offaly candidate – who now has 190 votes. As this is fewer than the two Laois candidates, the remaining Offaly candidate is eliminated and any votes that can be transferred are redistributed. But, with only two candidates left for two vacant seats, the two Laois Party candidates are elected by default.

Table: Hare-STV Election

Count 1Count 2Count 3
Offaly A340 (elected)20
Laois A226226226283 (elected)
Laois B224224224262 (elected)
Offaly B170170190

Not only are the two Laois Party candidates elected despite falling far short of the quota, but the Laois Party has also managed to take a majority of seats even though the majority of voters prefer the Offaly Party to the Laois Party. Such a result would not be possible under Droop-STV, where a party that is preferred by at least half of voters will always take at least half of seats.

Indeed, if we repeat the election under Droop-STV, where the quota would be 241 (960 voters / (3 seats + 1)+1), the larger surplus of the first Offaly candidate would, when transferred, elect the second Offaly candidate in the second count. The final seat would go to the Laois Party.

Table: Droop-STV Election

Count 1Count 2Count 3Count 4
Offaly A340 (elected)99
Laois A226226226226234464 (elected)
Laois B224224224224230
Offaly B170170269 (elected)28

No matter how you vote, remember to vote. Your vote is your voice and make sure it’s counted!!