Hare vs Droop: How to set the quota under STV

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 Single Transferable Vote). The group quickly settled on Hare’s system as the best option for electoral reform.

The Proportional Representation Society may have since become the Electoral Reform Society, but it still promotes the Single Transferable Vote – though with a few tweaks to Hare’s original ideas, many of them picked up from real-life implementations of the system.

With STV, each constituency elects a small group of MPs and voters rank candidates according to their preferences. To work out who gets elected, you first need to work out the quota. 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 extant preference. This continues until all the seats have been filled.

But how do you work out where the quota – the winning line – is? This is one of the things that has been tweaked since Hare’s original designs were laid out.

Hare vs Droop – Electoral Quotas

The two main electoral quotas 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 MPs 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 and it is the preferred option of the Electoral Reform Society, 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 ‘quirks’…

Firstly, there is an inequality issue. 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 Mountain Party and the Sea Party, with each running two candidates for three seats and 960 voters.

• 510 voters give their first preference to a Mountain Party candidate – 340 for the first candidate and 170 for the second candidate.
• 450 voters choose the Sea 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 Mountain candidate would be instantly elected, and their 20 surplus votes transferred to the other Mountain candidate – who now has 190 votes. As this is fewer than the two Sea candidates, the remaining Mountain candidate is eliminated and any votes that can be transferred are redistributed. But, with only two candidates left for two vacant seats, the two Sea Party candidates are elected by default.

Table: Hare-STV Election

Not only are the two Sea Party candidates elected despite falling far short of the quota, but the Sea Party has also managed to take a majority of seats even though the majority of voters prefer the Mountain Party to the Sea 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 Mountain candidate would, when transferred, elect the second Mountain candidate in the second count. The final seat would go to the Sea Party.

Table: Droop-STV Election

Of course, the relative downsides of Hare-STV are nothing like the major failures of the current First Past the Post system. Though examples like the above can happen, they are both a rarity and can be avoided by choosing Droop-STV instead. FPTP, however, regularly leads to dozens of seats across the country where the majority of voters would have preferred an MP from another party and has left roughly half of voters with an MP they had no hand in electing.

The move from Hare to Droop also shows the adaptability of STV and how it has been improved over the years, while FPTP remains the same failed voting system it was when the PRS was founded – still making the same mistakes, still failing to adequately represent British voters.