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

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 (called the Jefferson method in the USA), which is used to elect many of Europe’s national parliaments as well as the regional seats in Scotland and Wales. Counting takes place in rounds, with the party with the highest total in each round winning the seat. 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. To get a look at how D’Hondt works, let’s apply it to the votes cast in Nottinghamshire at the last general election.

D’Hondt, Nottinghamshire 2019

As the party with the most votes, the Conservatives would take the first seat. In the second round, the Conservative’s original vote tally has been divided by two (they have 1 seat, plus 1), Labour now has the highest tally so they win a seat. In the third round, Labour’s original votes are also now divided by two (they have 1 seat, plus 1) so the Conservatives return to having the highest total and take a second seat. In the next round, their original tally is divided by three (their 2 seats, plus one). 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, plus one. 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 Liberal Democrats to take a seat at the expense of Labour.

Table B: Sainte-Laguë, Nottinghamshire 2019

With the most votes, the Conservative win the first seat. In the second round, their original total has been divided by three (Double their seat count, plus one), and Labour now has the highest total and wins the next seat. This process continues until all the seats are full.

Comparing the Formulas

With Nottinghamshire, both electoral formulas produce different results but 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 both formulas.

The differences between the formulas are heightened when you have smaller constituencies and the differences that do appear add up across the country. Sainte-Laguë usually produces the most proportional results and D’Hondt is slightly less proportional on account of a moderate bias towards larger parties, though it is still vastly more proportional than FPTP.

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.