15 August 2022

In democratic elections, the number of seats in parliament is to be proportionate to the number of votes. But the mathematical methods that are used to calculate this today result in a somewhat skewed distribution. Now, an optimisation model has been developed at Linköping University, one which gives a better reflection of the will of the people.

Kaj Holmberg.
"But why use an old method when there is a better one – the best – at our disposal?", says Kaj Holmberg.  Magnus Johansson

Have you ever wondered what happens after an election? You might think that the votes are counted, and then the result is final. But it’s not quite that simple. First, the seats in parliament must be distributed between the parties that made it over the 4% minimum for the Swedish parliament. Specific mathematical methods are used in this process, to turn percentages into integers equivalent to parliamentary seats. But the method used today doesn’t give a perfectly proportionate result.

“Simply put, it’s not really fair. Many of the models used today to count seats after elections are over 100 years old. For example, the so-called “adjusted odd numbers method” was first proposed in 1910. It’s not based on optimisation, but instead classical mathematics”, says Kaj Holmberg, professor at Linköping University’s Department of Mathematics.

Old methods

In Sweden, it is the adjusted odd numbers method, also called the modified Sainte Laguë method, to calculate seat distribution. Similar methods are used in other countries. What these methods have in common is that they are old, and don't quite reflect the will of the people as well as they could. Taking the adjusted odd numbers method as an example, large parties are treated differently to smaller parties, and the 4% minimum threshold isn’t handled optimally.

“If you look at the Swedish general election of 2018, the result would have been different with optimal seat distribution. In that scenario, the so-called right-wing coalition would have won by one seat, instead of the left-wing coalition having the majority”, says Kaj Holmberg.

In the 2018 election, 6,476,725 valid votes were cast in ballot boxes around the country, determining how the 349 seats in parliament should be distributed. This means that each seat was equivalent to 18,557 votes. Or, expressed the other way round – each vote was equivalent to 0.0000538 seats. So if we divide the number of votes that each party received by the total amount of votes, then we get perfect proportions. In other words, the will of the people. But the percentages which result cannot be directly applied to parliamentary seats, which by necessity are only counted in whole numbers. In other words, you cannot have 7.84 parliamentary seats.

“The distribution of parliament’s seats should reflect the will of the people. Kaj Holmberg, portrait.Kaj Holmberg, professor the Department of Mathematics. Photo credit Magnus Johansson The problem is, it has to be a whole number. If you just round down or up, then you might end up with too many, or too few, seats in parliament. This research should not in any way be seen as a partisan intervention. It’s merely about using the best available method, one which we have now developed”, says Kaj Holmberg.

Problem before solution

He has developed a mathematical optimisation model and method that treats all parties equally, all votes equally, and which also takes proper account of the 4% minimum threshold. The difference between this and the method currently in use is that we have an exact understanding of the model in use.

“I always say to my students that you have to set up the optimisation problem before trying to find a solution. The current approach is to tinker around the edges of the existing method, instead of starting over, getting to the root of the problem, and then solving it correctly”, says Kaj Holmberg.

But the question is whether a new method will be implemented any time soon. Politics is a sensitive subject, and the feeling of tradition is strong. So Kaj Holmberg believes that the introduction of this method might be quite some time off.

“But why use an old method when there is a better one – the best – at our disposal? I hope that this discussion will now take place, and that there will be a willingness to use the better method in the long run."

Facts in brief: The model uses piecewise linearisation of the Euclidean distance between the number of votes and the distribution of seats, and can be used with a newly developed method of the same kind used today.

The article: Optimal proportional representation, Kaj Holmberg.



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