Mathematics shows that every electoral system must compromise among proportionality, local representation, and parliament size.
Denmark’s 2022 national election came down to a single seat, even though its proportional voting system was designed to make party representation closely match the vote. That outcome points to a deeper mathematical problem: no electoral system can always preserve local winners, guarantee proportional national results, and keep parliament at a fixed size.
Researchers from the University of Cambridge and the University of Copenhagen reached that conclusion after analyzing proportional and first-past-the-post systems. Their work shows that every method of electing a national legislature requires compromises, most often among regional representation, proportionality, and the number of parliamentary seats.
Those tensions have become increasingly visible as political landscapes across northern Europe have fragmented into more parties. Denmark’s left bloc won the 2022 election despite receiving fewer votes than the right bloc. Germany’s attempt to protect both local representation and proportionality caused the Bundestag to expand sharply before reforms were introduced in 2023. In the UK, Labour secured 63% of the seats with 33.7% of the vote in 2024, producing the country’s least proportional general election on record.
A perfectly ‘fair’ system may therefore be impossible, but the researchers argue that electoral rules can still be designed to balance these competing goals more effectively. Their findings were published in the Annals of Operations Research.
Denmark exposed the system’s limits
The investigation began after Denmark’s 2022 national election, when votes and seats failed to align for the first time in 75 years. Like many European democracies, Denmark uses a two-tier proportional system. Voters first choose local representatives, after which ‘leveling’ seats are distributed to bring each party’s share of parliament closer to its national vote share.
That mechanism becomes harder to maintain as more political parties compete. Greater fragmentation requires a larger number of ‘leveling’ seats to preserve proportionality, and the 40 available in Denmark’s 2022 election were no longer sufficient.
“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 seat more than their 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.”
Klausen and co-author Sebastian Tim Holdum of the University of Copenhagen used the election as the starting point for an impossibility theorem. Their result explains why growing party fragmentation is placing established voting systems under pressure and changing real democratic outcomes.
Three goals cannot always coexist
The researchers compared electoral systems used around the world, including the UK’s first past the post model and the proportional systems common across Europe. They identified three features that are often treated as essential to a ‘fair’ election:
- 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.”
The theorem shows that once enough parties are involved, an electoral system cannot guarantee all three conditions at once. Preserving two of them inevitably weakens the third.
Countries choose different compromises
Governments have responded by deciding which principle they are prepared to relax. Germany allowed the Bundestag to grow from 598 seats to 736 in 2021 as additional leveling seats were required to maintain proportionality.
Reforms adopted in 2023 stopped the chamber from expanding indefinitely. However, the change removed the guarantee that every local winner would enter parliament, prioritizing a fixed chamber size instead. Sweden uses a similar approach.
The UK makes a different trade-off. Its first-past-the-post system protects local winners and maintains a set number of seats, but it does not attempt to make national seat totals proportional to national vote shares.
That imbalance was less disruptive when British politics functioned largely as a two-party system. As additional parties have gained support, however, the difference between votes and seats has widened. Reform UK received 14.3% of the national vote in 2024 but secured only five of 650 seats, while Labour won 411 seats with 33.7% of the vote.
“In the next British election, the votes-to-seats mismatch could be even bigger,” said Klausen.
Several former British territories inherited the same system and have faced similar debates. Canada has retained first past the post despite repeated reform efforts. New Zealand adopted proportional representation in 1993, but its House of Representatives does not have a fixed number of seats.
A proportional fix sacrifices local wins
Klausen and Holdum also developed a possible mathematical alternative called ‘geographically ranked guaranteed proportionality.’ Their method would determine each party’s total number of seats from its national vote share before geography is considered. A party receiving 20% of the vote would therefore receive 20% of the seats.
Individual constituencies would then be ranked according to the strength of each candidate’s local result. Seats would be awarded by moving down that ranking until each party filled its national allocation.
The approach would guarantee proportionality, but it would weaken regional representation. A candidate could win comfortably in a constituency and still lose the seat if their party had already reached its quota through stronger results elsewhere.
“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 That either proportionality, regionality, or fixed parliament size must be sacrificed is no excuse. One can have all three principles approximately, yet many systems do not.”
Reference: “Impossibility theorem for two-tier electoral systems” by Sebastian Tim Holdum, and Frederik Ravn Klausen, 31 July 2026, Annals of Operations Research.
DOI: 10.1007/s10479-026-07344-1
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6 Comments
Considering that these various electoral procedures were developed by what are intelligent, experienced people, the fact that they don’t behave as desired or expected should give one pause. That is, should one consider whether the problem of not working as expected be written off as a fluke, or should one consider an alternative hypothesis? That is, might the anomalous behavior be attributed to what is assumed to be a ‘fair coin’ but is actually not a ‘fair coin?’ That is, perhaps the system is being gamed fraudulently! I think that hypothesis should be considered seriously, even if the experts claim it is improbable. Sherlock Holmes said, “Eliminate all other factors, and the one which remains must be the truth.”
This is Patriots want a small government with the lowest taxes and complete privatization of education, medical care, and social security.
Administrative tasks must be transferred en masse from the national level back to the municipal, provincial, or regional level.
Citizens will be involved in regular general administrative meetings.
But first and foremost, the European Union must be abolished and replaced by the European Economic Community based on bilateral treaties.fundamentally false.
It’s all a simple fix with cheating – and cheating seems to have been an inherent aspect.
Everyone wants something at the highest value at the lowest cost, without question. Subterfuge and cheating subtend the evolution of intelligence. Systems of corruption abound throughout recorded history, in extreme, reverting to conflict and social destruction. Humans can rarely find means to agree, and revert to disagreeable solutions, and they are driven by biological imperatives which can not be resisted, or even recognized.
“Humans can rarely find means to agree, and revert to disagreeable solutions, …”
I think that you are being pessimistic using the word “rarely.” Certainly, disagreements occur. But the world’s second oldest profession is evidence that people have been concerned about the problem and have sought institutional solutions since the dawn of civilization. Laws, lawyers, and courts are an attempt to provide an alternative to violence. Even democracy is an attempt to provide a contractual alternative to revolution and even wars of conquest. That is why the United Nations was created. While our attempts sometimes fail, the facts that vigilantism and anarchy aren’t the dominant behaviors of people suggests that our Rule of Law works most of the time. The biological imperatives can obviously be recognized or else we wouldn’t be discussing it. I am concerned, however, about the shortsightedness of some people who are unwilling to abide by the fundamental social contract of democracy. Namely, that it only works if everyone accepts the outcome of an election and don’t try to overturn the results or obstruct the platforms of the newly-elected party. Perhaps we can attribute the latter problem to stupidity, which can’t be fixed.
“The mathematical barrier in democratic design—Arrow’s Impossibility Theorem—isn’t an unsolvable human paradox; it’s a failure of single-dialect translation. Traditional voting forces voters to convert nuanced preferences into a rigid rank-order list (1st, 2nd, 3rd), stripping out intensity and forcing defensive, strategic voting that maximizes polarization. “The primary obstacle to modernizing voting mathematics isn’t the theory—it’s the incumbency paradox: the lawmakers empowered to change the rules are the direct beneficiaries of the system that elected them.
Expecting a partisan legislature to voluntarily adopt consensus-based cardinal voting (like Majority Judgment or Score Voting) is an operational dead end. Real structural reform requires shifting the boundary conditions around them: Instead of asking “Who do you want?”, the ballot asks: “What level of merit or support do you give each candidate?”Candidate as Excellent , Very Good , Acceptable , Poor , Reject . Why this changes your choices:No “Spoiler” Effect: You don’t have to vote for a candidate you dislike just to stop a worse one. You can give an “Excellent” rating to an underdog third-party candidate and a “Very Good” rating to a mainstream backup candidate at the same time.True Expression: You express your exact level of satisfaction for every option on the ballot . The Output Language (How the Winner Is Calculated)
In Standard voting (Plurality/Ranked Choice), votes are tallied by counting top choices and eliminating candidates—a process that creates “spoiler candidates” and strategic lying.
In a dual-language system, the algorithm calculates the Majority Grade (The Median Rating) for each candidate across the entire electorate. The algorithm Finds the Median: The winner is the candidate who receives the highest median grade from the entire population.
Break Ties via Consensus: If two candidates share the same median grade, the algorithm looks at which candidate has a higher percentage of ratings above that median grade (or lower percentage below it).