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Mathematical Proof Shows Perfectly Fair Elections Are Impossible — AI-generated representative image: A parliamentary chamber with electoral materials illustrating the mathematical impossibility of perfectly fair election syste

Editorial Team

Cambridge and Copenhagen researchers prove rising political fragmentation forces every electoral system to sacrifice local representation, national proportionality, or fixed parliament size

Researchers from the University of Cambridge and the University of Copenhagen have published a mathematical proof demonstrating that no electoral system can simultaneously guarantee local representation, national proportionality, and a fixed parliament size when multiple political parties compete. The findings, published in the Annals of Operations Research, show that every method for electing a national legislature involves unavoidable compromises among these three principles.

The study arrives as electoral systems across Europe and beyond face mounting strain from rising political fragmentation. With voters increasingly dividing support among a growing number of parties, the mathematical tensions identified by the researchers are producing visible consequences, from distorted election outcomes to ballooning parliaments. The findings carry particular relevance for the UK, where debates over proportional representation have intensified following the country's least proportional election result on record in 2024.

Mathematical Constraints Identified

The researchers, Dr Frederik Ravn Klausen of Cambridge and Sebastian Tim Holdum of the University of Copenhagen, developed an impossibility theorem identifying three features that cannot all be preserved once a certain number of political parties are in play:

Regionality: every candidate who wins a local constituency keeps their seat
Proportionality: national seat totals match national vote shares
Fixed parliament size: the number of seats remains unchanged between elections

"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."

Real-World Consequences Across Europe

The mathematical tensions identified in the study have already produced measurable effects in several countries. Denmark's 2022 election marked the first time in 75 years that votes and seats failed to correspond under its two-tier proportional system. The left bloc won despite receiving fewer votes than the right bloc because the 40 leveling seats available were insufficient to correct the imbalance. "Essentially, it almost broke the system," Klausen said. "The end result was that the Social Democrats got one seat more than their votes corresponded to."

Germany experienced a different strain: the Bundestag expanded from 598 seats to 736 seats by 2021 as the number of leveling seats required to maintain proportionality continued to grow. Reforms enacted in 2023 capped the expansion but removed the guarantee that every local constituency winner would enter parliament. Sweden has adopted a similar approach.

The UK's Extreme Trade-Off

The United Kingdom's first-past-the-post system makes a different compromise by sacrificing proportionality. The 2024 general election produced the country's most disproportionate result on record: Labour secured 63% of parliamentary seats with 33.7% of the vote, while Reform UK received 14.3% of the national vote but won only five of the 650 seats. "In the next British election, the votes-to-seats mismatch could be even bigger," Klausen warned.

Several former British territories face similar pressures. Canada has retained first-past-the-post despite repeated reform efforts, while New Zealand adopted proportional representation in 1993, though its parliament does not have a fixed size.

A Proposed Alternative

The researchers have also proposed a new algorithm called "geographically ranked guaranteed proportionality." Under this system, each party's national seat total would be determined entirely by its overall vote share, with constituencies ranked by local candidate performance and seats awarded from the top down until each party fills its allocation. The method guarantees proportionality but weakens regionality, as a locally popular candidate could still fail to enter parliament if their party's quota was already filled elsewhere.

"Our results show that as discussions around proportional representation in the UK get more attention, there is no such thing as a 'perfect' system," Klausen said. "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."