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

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Guest Author,

Posted on the 27th July 2021

This is a guest post from Dylan Difford who has recently completed an MA in Politics at the University of Essex, focussing on party and voting systems in Britain and Europe.

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

Con Lab LD Brexit Ash Ind Green Others
Votes 258,794 204,011 33,604 15,728 13,498 10,375 9,743
Round 1 258,794 204,011 33,604 15,728 13,498 10,375 9,743 1 Con
Round 2 129,397 204,011 33,604 15,728 13,498 10,375 9,743 1 Lab
Round 3 129,397 102,006 33,604 15,728 13,498 10,375 9,743 2 Con
Round 4 86,265 102,006 33,604 15,728 13,498 10,375 9,743 2 Lab
Round 5 86,265 68,004 33,604 15,728 13,498 10,375 9,743 3 Con
Round 6 64,699 68,004 33,604 15,728 13,498 10,375 9,743 3 Lab
Round 7 64,699 51,003 33,604 15,728 13,498 10,375 9,743 4 Con
Round 8 51,759 51,003 33,604 15,728 13,498 10,375 9,743 5 Con
Round 9 43,132 51,003 33,604 15,728 13,498 10,375 9,743 4 Lab
Round 10 43,132 40,802 33,604 15,728 13,498 10,375 9,743 6 Con
Round 11 36,971 40,802 33,604 15,728 13,498 10,375 9,743 5 Lab
Elected 6 5

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

Con Lab LD Brexit Ash Ind Green Others
Votes 258,794 204,011 33,604 15,728 13,498 10,375 9,743
Round 1 258,794 204,011 33,604 15,728 13,498 10,375 9,743 1 Con
Round 2 86,265 204,011 33,604 15,728 13,498 10,375 9,743 1 Lab
Round 3 86,265 68,004 33,604 15,728 13,498 10,375 9,743 2 Con
Round 4 51,759 68,004 33,604 15,728 13,498 10,375 9,743 2 Lab
Round 5 51,759 40,802 33,604 15,728 13,498 10,375 9,743 3 Con
Round 6 36,971 40,802 33,604 15,728 13,498 10,375 9,743 3 Lab
Round 7 36,971 29,144 33,604 15,728 13,498 10,375 9,743 4 Con
Round 8 28,755 29,144 33,604 15,728 13,498 10,375 9,743 1 LD
Round 9 28,755 29,144 11,201 15,728 13,498 10,375 9,743 4 Lab
Round 10 28,755 22,668 11,201 15,728 13,498 10,375 9,743 5 Con
Round 11 23,527 22,668 11,201 15,728 13,498 10,375 9,743 6 Con
Elected 6 4 1

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

Con Lab LD Brexit Ash Ind Green Others
Votes 258,794 204,011 33,604 15,728 13,498 10,375 9,743
Round 1 258,794 204,011 33,604 15,728 13,498 10,375 9,743 1 Con
Round 2 209,180 204,011 33,604 15,728 13,498 10,375 9,743 2 Con
Round 3 159,566 204,011 33,604 15,728 13,498 10,375 9,743 1 Lab
Round 4 159,566 154,397 33,604 15,728 13,498 10,375 9,743 3 Con
Round 5 109,952 154,397 33,604 15,728 13,498 10,375 9,743 2 Lab
Round 6 109,952 104,783 33,604 15,728 13,498 10,375 9,743 4 Con
Round 7 60,338 104,783 33,604 15,728 13,498 10,375 9,743 3 Lab
Round 8 60,338 55,169 33,604 15,728 13,498 10,375 9,743 5 Con
Round 9 10,724 55,169 33,604 15,728 13,498 10,375 9,743 4 Lab
Round 10 10,724 5,555 33,604 15,728 13,498 10,375 9,743 1 LD
Round 11 10,724 5,555 0 15,728 13,498 10,375 9,743 1 Brexit
Elected 5 4 1 1

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

Votes % Vote FPTP Seats D’Hondt Seats Sainte-Laguë Seats Hare-LR Seats
Con 258,794 47.4% 8 72.7% 6 54.5% 6 54.5% 5 45.5%
Lab 204,011 37.4% 3 27.3% 5 45.5% 4 36.4% 4 36.4%
LD 33,604 6.2% 1 9.1% 1 9.1%
Brexit Party 15,728 2.9% 1 9.1%
Ashfield Ind 13,498 2.5%
Green 10,375 1.9%
Others 9,743 1.8%
Deviation 25.3% 15.2% 10.1% 9.1%

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. It is also possible in certain circumstances for Hare-LR, and to a lesser extent Sainte-Laguë, to award less than half of seats to a party that has narrowly won more than of votes.

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.

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