Mathematicians have shown that completely fair elections are—in a sense—impossible. No electoral system can guarantee both that local votes translate directly into local seats and that total seat distribution matches overall vote shares—not without an indefinitely growing parliament.
The researchers, from the University of Cambridge and the University of Copenhagen, analyzed different electoral systems, both proportional and first-past-the-post, and found that in any system, there are always going to be a series of trade-offs when electing a national legislature. For most systems, these trade-offs are between regional representation, guaranteed proportionality and parliamentary size.
In the past decade, voters in northern European democracies have seen the results of these trade-offs, driven by the increasing fragmentation of political parties. In Denmark, the left bloc won the 2022 election despite winning fewer votes than the right bloc, even though Denmark has a proportional system. In Germany, the desire to balance regional representation and proportionality led to the Bundestag ballooning in size before reforms were made in 2023. And in the UK, the 2024 general election was its least proportional ever, as Labour won 63% of the seats with 33.7% of the vote.
However, while there may not be a completely "fair" electoral system, the researchers say there are ways to get around the tension between regionality, proportionality and parliament size and make the impossible possible. Their results are reported in the Annals of Operations Research.
The researchers began their study after the 2022 Danish national election, when the number of votes did not correspond to the number of seats for the first time in 75 years. The Danish system, like many in Europe, is a two-tier proportional system. Voters elect their local representatives, and then a number of "leveling" seats are added so that each party's seat share roughly matches its vote share. But as party fragmentation increases, the number of "leveling" seats also needs to increase to maintain that proportionality. In the 2022 Danish election, the 40 leveling seats were no longer enough to achieve proportionality.
"Essentially, it almost broke the system," said study co-author Dr. Frederik Ravn Klausen, from Cambridge's Department of Pure Mathematics and Mathematical Statistics. "The end result was that the Social Democrats got one more seat than its votes corresponded to. Of course, that is nothing by UK standards, but in a two-tier system, that isn't meant to happen. It increased the drama that this single seat decided the election."
Working with his co-author Sebastian Tim Holdum from the University of Copenhagen, Klausen formulated an impossibility theorem that illustrates how different voting systems are straining under increased fragmentation, which is having real effects on democratic outcomes.
The researchers looked at different electoral systems around the world, both first-past-the-post systems like the UK's and proportional systems like those in much of the rest of Europe. They identified three main components of a "fair" system:
- Regionality: Everyone who wins a local seat keeps it.
- Proportionality: National seat totals match national vote shares.
- Fixed parliament size: The number of seats doesn't change from election to election.
"We found that once there are a certain number of political parties, one of these three components has to give; otherwise, the math simply doesn't work," said Klausen. "It's not corruption or conspiracy, it's simply mathematics."
Different countries have made different choices when it comes to which of the three principles they're willing to sacrifice. In Germany, the Bundestag grew from 598 seats to 736 seats in 2021 because the required number of leveling seats kept growing. In 2023, the German system was reformed to prevent an ever-expanding Bundestag, but it sacrificed the guarantee of a local representative to keep the chamber a fixed size. Sweden is among the countries that have adopted a similar model.
In the UK, which has a first-past-the-post system, proportionality is the component that gets sacrificed. For most of the last century, the UK has been, in essence, a two-party system, but as new parties form and grow across the political spectrum, it leads to "unfair" results. In 2024, Reform UK won 14.3% of the vote but won just five of 650 seats, while Labour won 411 seats on 33.7% of the vote. "In the next British election, the votes-to-seats mismatch could be even bigger," said Klausen.
Many former British territories inherited the British system and are also struggling to adapt. Canada has kept the first-past-the-post system through repeated attempts at reform. New Zealand switched to proportional representation in 1993—but the New Zealand House of Representatives does not have a fixed parliament size.
However, Klausen says there could be a potential solution, also backed up by math. He and Holdum have proposed an algorithm they call "geographically ranked guaranteed proportionality," which could get around the impossibility of current systems. In their proposal, national totals are set before any voting takes place and irrespective of geography, so that any party that gets 20% of the vote gets 20% of the seats. Next, every individual seat is ranked by strength of local performance, and seats are allocated by working down that list.
However, this system, while proportional, does sacrifice regionality because a candidate could win their seat comfortably, but if their party has already used up its quota on stronger wins elsewhere, that candidate would not get to keep their seat.
"Our results show that as discussions around proportional representation in the UK get more attention, there is no such thing as a 'perfect' system," said Klausen. "However, one can still say that some systems perform better than others. The fact that either proportionality, regionality or fixed parliament size must be sacrificed is no excuse. All three principles can be achieved approximately, yet many systems do not."
More information
Impossibility theorem for two-tier electoral systems, Annals of Operations Research (2026). DOI: 10.1007/s10479-026-07344-1
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Citation: Math shows why your vote may not count the way you think it does (2026, July 31) retrieved 31 July 2026 from https://phys.org/news/2026-07-math-vote.html
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