What is the difference between D’hondt, Sainte-Laguë and Hare?

Looking at just Party List systems in western Europe, there is a lot of variation. We can see constituencies vary from four- or five-member districts to single nationwide constituencies, there are some countries that impose electoral thresholds and some where you vote for individuals as well as parties. But one of the most significant differences, is also the least visible – the equations that decide how the seats are actually allocated to the parties. So, just how do these electoral formulas work?

The D’Hondt Method

The most common electoral formula is the D’Hondt method, which is used to elect many of Europe’s national parliaments as well as the regional seats in Scotland and Wales. D’Hondt works by dividing the number of votes cast for each party by the number of seats they have already won, plus one – so that after a party has won one seat their votes are divided by two, after they have won two seats their votes are divided by three, and so on. Counting takes place in rounds, with the party with the highest total in each round winning the seat.

To get a look at how D’Hondt works, let’s apply it to the votes cast in Nottinghamshire at the last general election.

Table A: D’Hondt, Nottinghamshire 2019

As the party with the most votes, the Conservatives would take the first seat, with their vote tally then being divided by two. In the second round, Labour now has the highest tally and so they win a seat, and their votes are divided by two. In the third round, the Conservatives return to having the highest total and take a second seat, so their original tally is divided by three. This continues until all the seats are filled, with the Conservatives winning six seats and Labour five.

The Sainte-Laguë Method

D’Hondt’s biggest competitor is the Sainte-Laguë method, used in countries such as Germany, New Zealand and Sweden. Sainte-Laguë works in much the same way as D’Hondt, though the votes are divided by twice the number of seats won, add one – making the divisors 1, 3, 5, etc. rather than 1, 2, 3, etc. This has the effect of slightly improving proportionality between parties and being more favourable to smaller parties. Applying Sainte-Laguë to Nottinghamshire would allow the Lib Dems to take a seat at the expense of Labour.

Table B: Sainte-Laguë, Nottinghamshire 2019

The Hare-Largest Remainder Method

The only other electoral formula that is commonly used is the Hare-Largest Remainder (Hare-LR) method. Unlike D’Hondt or Sainte-Laguë, Hare-LR is quota-based rather than divisor-based. This means that parties win seats based on how many times the exceed the Hare quota – which is total votes divided by total seats. As this won’t fill all the seats, those that are left are allocated to the parties with the most remaining votes. If we return to Nottinghamshire, where the Hare quota would be set at 49,614, we can see another different allocation – with the Brexit Party taking the final seat.

Table C: Hare-LR, Nottinghamshire 2019

In each round, the party with the most votes loses a quota’s worth of votes (49,614 in this case) and gains an MP. 

Comparing the Formulas

With Nottinghamshire, all three electoral formulas produce different results and are, unsurprisingly, all more proportional than the actual First Past the Post result. However, different allocations aren’t guaranteed and, if we applied the same process to Cheshire, we would get the same results under all three formulas. But the differences that do appear do add up across the country.

The differences between the formulas are heightened when you have smaller constituencies. Hare-LR usually produces the most proportional results, with Sainte-Laguë not that far behind. D’Hondt is slightly less proportional on account of a moderate bias towards larger parties, though it is still vastly more proportional than FPTP.

While Hare-LR has higher levels of proportionality it is sometimes seen as unfair to award seats to parties that have won a low share of the quota – such as the Brexit Party in our Nottinghamshire example.

Just as every electoral system is a compromise between proportionality, voter choice and local representation, there is no simple way to declare one electoral formula the best. But they all are an improvement on Westminster’s broken First Past the Post system.