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Maths shows why your vote may not count the way you think it does

Mathematicians have shown that completely fair elections are – in a sense – impossible. No electoral system can guarantee both that local votes translate directly into local seats, and that total seat distribution matches overall vote shares: not without an indefinitely growing parliament.

The researchers, from the University of Cambridge and the University of Copenhagen, analysed different electoral systems, both proportional and first past the post, and found that in any system, there are always going to be a series of trade-offs when electing a national legislature. For most systems, these trade-offs are between regional representation, guaranteed proportionality and parliamentary size.<br>In the past decade, voters in northern European democracies have seen the results of these trade-offs, driven by the increasing fragmentation of political parties. In Denmark, the left bloc won the 2022 election despite winning fewer votes than the right bloc, even though Denmark has a proportional system. In Germany, the desire to balance regional representation and proportionality led to the Bundestag ballooning in size, before reforms were made in 2023. And in the UK, the 2024 general election was its least proportional ever, as Labour won 63% of the seats with 33.7% of the vote.<br>However, while there may not be a completely ‘fair’ electoral system, the researchers say there are ways to get around the tension between regionality, proportionality and parliament size, and make the impossible, possible. Their results are reported in the Annals of Operations Research.<br>The researchers began their study after the 2022 Danish national election, where the number of votes did not correspond to the number of seats for the first time in 75 years. The Danish system, like many in Europe, is a two-tier or proportional system. Voters elect their local representatives, and then a number of ‘levelling’ seats are added so that each party’s seat share roughly matches its vote share. But as party fragmentation increases, the number of ‘levelling’ seats also needs to increase to maintain that proportionality. In the 2022 Danish election, the 40 levelling seats were no longer enough to achieve proportionality.<br>“Essentially, it almost broke the system,” said study co-author Dr Frederik Ravn Klausen, from Cambridge’s Department of Pure Mathematics and Mathematical Statistics. “The end result was that the Social Democrats got one seat more than their votes corresponded to. Of course that is nothing by UK standards, but in a two-tier system, that isn’t meant to happen. It increased the drama that this single seat decided the election.”<br>Working with his co-author Sebastian Tim Holdum from the University of Copenhagen, Klausen formulated an impossibility theorem that illustrates how different voting systems are straining under increased fragmentation, which is having real effects on democratic outcomes.<br>The researchers looked at different electoral systems around the world, both first-past-the-post like the UK, and proportional systems like much of the rest of Europe. They identified three main components of a ‘fair’ system:<br>Regionality: everyone who wins a local seat keeps it<br>Proportionality: national seat totals match national vote shares<br>Fixed parliament size: the number of seats doesn’t change from election to election<br>“We found that once there are a certain number of political parties, one of these three components has to give, otherwise the maths simply don’t work,” said Klausen. “It’s not corruption or conspiracy, it’s simply mathematics.”<br>Different countries have made different choices when it comes to which of the three principles they’re willing to sacrifice. In Germany, the Bundestag grew from 598 seats to 736 seats in...