Researchers from the University of Cambridge and the University of Copenhagen have demonstrated mathematically that perfectly fair election systems are impossible. Their analysis, published in the Annals of Operations Research, shows that every method for electing a national legislature must compromise among three key principles: ensuring local victories produce local seats, making national seat totals accurately reflect overall vote shares, and maintaining a fixed parliament size.
The study emerged from observations in recent northern European elections that highlighted these inherent tensions. In Denmark’s 2022 election, the left bloc won despite receiving fewer votes than the right bloc, despite the country using proportional representation. Germany expanded its Bundestag significantly before introducing reforms in 2023 to address the rising number of leveling seats needed for proportionality. The UK’s 2024 general election produced the country’s least proportional result on record, with Labour winning 63% of seats while receiving 33.7% of the vote.
The researchers identified that as political support fragments across more parties, electoral systems face mounting strain. Denmark’s system, which uses a two-tier approach with leveling seats to preserve proportionality, failed for the first time in 75 years during the 2022 election. The problem stems from the mathematical reality that once a certain number of political parties emerge, at least one of the three core fairness principles must be compromised.
Different countries have responded by sacrificing different principles. Germany chose to expand parliament size while weakening the guarantee that local winners would enter parliament. The UK prioritizes local representation but sacrifices proportionality under its first-past-the-post system. The researchers proposed an alternative algorithm called “geographically ranked guaranteed proportionality” that would guarantee proportional seat allocation while weakening regional representation.
While the study confirms that mathematical limits cannot be eliminated, the researchers argue that countries can still design systems that balance the competing goals more effectively than current approaches. According to the findings, though no perfect system exists, some electoral designs perform substantially better than others at approximating all three principles simultaneously.
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