Two Mathematicians Just Proved That a Perfectly Fair Election Is Impossible

Two mathematicians have proved that no electoral system can be fair in all the ways voters expect it to be — not as a matter of political will, but as a matter of arithmetic.

Writing in the Annals of Operations Research, Frederik Ravn Klausen of the University of Cambridge and Sebastian Tim Holdum of the University of Copenhagen show that three properties every modern democracy claims to want become mutually incompatible once a country's politics fragment past a certain number of parties.

The three properties are easy to state. Regionality: whoever wins a local district actually gets that seat. Proportionality: a party's share of seats in the national legislature matches its share of the national vote. Fixed size: the legislature has a set number of seats. Each of those sounds like a minimum requirement rather than a luxury. The proof shows you can have any two.

"Once there are a certain number of political parties, one of these three components has to give, otherwise the maths simply don't work," Klausen said.

The result is not abstract, and the authors point to cases where a real country has already had to choose which principle to sacrifice. Germany protects proportionality and lets the chamber grow: the Bundestag swelled from a nominal 598 seats to 736 in order to compensate parties whose national vote share exceeded their district wins. Denmark holds the size fixed and accepts distortion — in 2022 the left bloc took a governing majority despite receiving fewer total votes than the right. The United Kingdom protects regionality and a fixed size and abandons proportionality altogether: in 2024 Labour won 411 seats on 33.7% of the vote, while Reform UK took 14.3% of the vote and five seats.

The work sits in the tradition of Kenneth Arrow's 1951 impossibility theorem, which showed that no ranked voting rule can convert individual preferences into a coherent group preference while satisfying a short list of reasonable conditions. Arrow's result was about how ballots are aggregated. Klausen and Holdum's is about geography — the fact that legislatures are built out of places, and places do not divide neatly along party lines.

Rather than stopping at the impossibility, the two proposed a specific compromise they call geographically ranked guaranteed proportionality. It fixes the size of the chamber and guarantees national proportionality exactly, and it pays for both by weakening regionality in a controlled way: a candidate who wins their district can still be passed over if their party has already filled its national quota, with the seat going instead to a strong performer from a party still owed representation. The tradeoff is explicit and rule-based rather than emergent and arbitrary, which is the authors' central argument for it.

The timing is not incidental. Redistricting fights are running in several U.S. states ahead of the November midterms, Germany has spent two election cycles trying to shrink its Bundestag back toward its nominal size, and proportional-representation campaigns in Britain and Canada have gained ground since 2024. The paper does not tell any of them what to do. It tells them that whatever they choose, something has to break.