Mathematicians Show No Voting System Can Perfectly Balance All Fairness Criteria

A joint team from the University of Cambridge and the University of Copenhagen has mathematically proven that a perfectly fair election—one that preserves local victories, reflects national vote shares and keeps parliament size unchanged—is impossible once a sufficient number of parties contest. Their analysis draws on recent elections in Denmark, Germany, the United Kingdom and elsewhere, illustrating how each system compromises at least one of these principles.

The researchers outline how Germany expanded its Bundestag, while the UK’s first-past-the-post system sacrifices proportionality. To mitigate the dilemma, they introduce a voting scheme named geographically ranked guaranteed proportionality, which guarantees proportional seat allocation but weakens regional representation. The work, published in the Annals of Operations Research, suggests that while mathematical limits cannot be removed, electoral designs can be improved to balance the competing aims more effectively.