Mathematicians prove perfectly fair elections impossible

Mathematicians proved on August 2, 2026, that no electoral system can simultaneously guarantee local representation, national proportionality, and a fixed parliament size. The finding, published in the Annals of Operations Research, affects every democracy that elects a national legislature.

Dr. Frederik Ravn Klausen of the University of Cambridge and Sebastian Tim Holdum of the University of Copenhagen authored the study. Their impossibility theorem stems not from flawed design but from inherent mathematical constraints.

The research was inspired by Denmark’s 2022 election, where the left bloc won despite fewer votes because 40 levelling seats proved insufficient for full proportionality. Klausen noted that the Social Democrats received one seat more than their vote share warranted, deciding the government.

As political fragmentation grows, such tensions become more visible. Germany’s Bundestag swelled from 598 to 736 seats in 2021 to maintain proportionality before 2023 reforms stripped the guarantee that every local winner enters parliament. Britain’s 2024 election delivered Labour 63% of seats from 33.7% of the vote, while Reform UK’s 14.3% share yielded only five of 650 seats.

The authors propose an alternative algorithm, ‘geographically ranked guaranteed proportionality’, which awards parties seats solely by national vote share and then ranks constituencies by local performance. The method ensures proportionality but weakens regionality: a local winner might be excluded if their party’s quota is already filled.

Klausen said that while no perfect system exists, countries can nonetheless design frameworks that balance local representation, proportionality, and fixed size more effectively.