Cambridge and Copenhagen mathematicians prove three-way fair-election trade-off

Mathematicians at the University of Cambridge and the University of Copenhagen conclude that no electoral method can always deliver local seat retention, national proportionality, and a fixed parliament size at once, ScienceDaily reported from Cambridge materials published in the Annals of Operations Research. The authors identify three features many people treat as fair: regionality, meaning everyone who wins a local seat keeps it; proportionality, meaning national seats match national vote shares; and a fixed parliament size. Once there are enough political parties, one of the three must give, they found. The project followed Denmark's 2022 election, when a two-tier proportional system with 40 leveling seats could not fully restore proportionality for the first time in 75 years. Co-author Dr Frederik Ravn Klausen said the Social Democrats received one seat more than their votes corresponded to, and that single seat decided the election. ScienceDaily said Germany expanded the Bundestag from 598 to 736 seats in 2021 to preserve local representation and proportionality before 2023 reforms, while the UK's first-past-the-post system often sacrifices proportionality. In 2024, Reform UK received 14.3% of the national vote but five of 650 seats, and Labour won 411 seats with 33.7% of the vote, according to the write-up. The paper is titled Impossibility theorem for two-tier electoral systems and carries DOI: 10.1007/s10479-026-07344-1.