Mathematicians have shown that a perfectly fair election system is impossible — in a precise mathematical sense. No voting method can always guarantee that local victories produce local seats, that national seat totals accurately reflect overall vote shares, and that the size of parliament stays fixed.
Researchers from the University of Cambridge and the University of Copenhagen analyzed proportional representation and first-past-the-post systems and found that every method for electing a national legislature involves compromises, usually among three principles: regionality (everyone who wins a local seat keeps it), proportionality (seat totals match vote shares), and a fixed number of seats.
"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," says Dr. Frederik Ravn Klausen of Cambridge's Department of Pure Mathematics and Mathematical Statistics. "It's not corruption or conspiracy, it's simply mathematics."
The study, published in the Annals of Operations Research, was inspired by Denmark's 2022 election, when votes and seats failed to correspond for the first time in 75 years. Denmark uses a two-tier system with "leveling" seats meant to correct proportionality — but as support spreads across more parties, more leveling seats are needed than are available.
The researchers point to real-world examples of the tension. Germany's Bundestag swelled from 598 to 736 seats in 2021 before 2023 reforms stopped the expansion — but the reforms also removed the guarantee that every local winner enters parliament. In the UK, the 2024 general election was the least proportional on record: Labour won 63% of seats with 33.7% of the vote, while Reform UK received 14.3% of the vote and only five of 650 seats.
Klausen and co-author Sebastian Tim Holdum also propose a new algorithm called "geographically ranked guaranteed proportionality," under which each party's national seat total is determined entirely by its vote share, with constituencies ranked by how strongly candidates performed locally. The method guarantees proportionality but weakens regionality — a candidate could win a constituency and still miss parliament if their party's quota is already filled.
"Our results show that there is no such thing as a 'perfect' system," says Klausen. "However, one can still say that some systems perform better than others. That either proportionality, regionality, or fixed parliament size must be sacrificed is no excuse."
