Researchers from Cambridge and Copenhagen proved an impossibility theorem for two-tier electoral systems and proposed an algorithm that approximately balances all three goals.

Most voters assume that an election turns votes into seats in a fair and predictable way. A new mathematical analysis suggests that, in many systems, something has to give.

Writing in Annals of Operations Research, Frederik Ravn Klausen of the University of Cambridge and Sebastian Tim Holdum of the University of Copenhagen present what they call an impossibility theorem for two-tier electoral systems. These systems combine local constituencies, in which voters elect their own representatives, with a national layer meant to make each party’s share of seats match its share of the vote.

The researchers show that no such system can fully guarantee three things at once: that every local winner keeps their seat, that national seat totals match vote shares, and that the size of parliament stays fixed. Satisfying all three would require a parliament that keeps growing.

“We found that once there are a certain number of political parties, one of these three components has to give,” said Klausen, of Cambridge’s Department of Pure Mathematics and Mathematical Statistics. Otherwise, he added, “the maths simply don’t work.”

The Danish surprise

The project was prompted in part by Denmark’s 2022 general election. Denmark uses a two-tier system in which 40 levelling seats, also called compensatory seats, are distributed after the constituency results to bring each party’s total in line with its national vote share.

In 2022, for the first time in 75 years, that mechanism fell short. The 40 levelling seats were not enough to correct the imbalance created by constituency results. The left bloc won the election despite receiving fewer votes than the right bloc.

“Essentially, it almost broke the system,” Klausen said. “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,” he added.

Three goals, pick two

The authors examined both proportional systems and first-past-the-post systems, in which the candidate with the most votes in each constituency wins and there is no national correction. They found that most systems trade off regional representation, proportionality and parliament size in different ways.

Germany illustrates one trade-off. Its two-tier system long guaranteed both a local representative for every constituency and proportional results, which it achieved by adding extra seats as needed. As a result, the Bundestag grew from 598 seats to 736 in 2021. A 2023 reform capped its size, but at the cost of ending the guarantee that every constituency winner would take a seat. Sweden uses a similar model.

The growth came from so-called overhang seats. When a party won more constituency seats in a state than its share of the party-list vote entitled it to, it kept the extra seats, and other parties then received additional levelling seats to restore proportionality. As more parties entered the Bundestag and voters split their ballots more often, the number of extra seats ballooned.

First-past-the-post systems make a different sacrifice. In the United Kingdom’s 2024 general election, Labour won 411 of 650 seats, or 63 percent, with 33.7 percent of the vote. Reform UK won 14.3 percent of the vote but only five seats. The authors describe it as the least proportional result in UK history.

Other countries have chosen different paths. New Zealand moved to a proportional system after a 1993 referendum but does not fix the size of its parliament. Canada has kept first past the post despite repeated attempts at reform.

“It’s not corruption or conspiracy, it’s simply mathematics,” Klausen said.

A proposed compromise

The researchers do not conclude that all systems are equally flawed. They propose an algorithm they call geographically ranked guaranteed proportionality.

Under the scheme, each party’s national seat total is set by its share of the vote, so a party with 20 percent of the vote would receive 20 percent of the seats. Those seats are then allocated to candidates by ranking constituencies according to how strongly the party performed in each. The parliament stays a fixed size and remains proportional, while seats still go mostly to candidates with strong local support.

The trade-off is that a candidate who wins locally may not take a seat if the party’s quota has already been filled by stronger results elsewhere. In that sense, the method relaxes the guarantee of local winners while keeping the other two goals intact.

The method shares some features with Germany’s 2023 reform, under which constituency winners whose party has not earned enough seats through its share of the vote may not take their seats. The difference is that geographically ranked guaranteed proportionality applies the ranking systematically across the country, so the seats a party does receive go to its strongest local results.

Lessons for reform debates

The findings come as electoral reform draws renewed attention in several countries. In the UK, the gap between votes and seats in 2024 has fueled discussion of proportional representation, and Klausen warned that the mismatch could grow. “In the next British election, the votes to seats mismatch could be even bigger,” he said.

As those discussions continue, Klausen said, the research shows that “there is no such thing as a ‘perfect’ system. However, one can still say that some systems perform better than others.”

“That either proportionality, regionality and fixed parliament size must be sacrificed is no excuse,” he said. “One can have all three principles approximately, yet many systems do not.”

The result echoes a famous line of work in social choice theory. In the 1950s, the economist Kenneth Arrow proved that no ranked voting method can satisfy a set of seemingly reasonable fairness conditions simultaneously, a finding that helped earn him a Nobel Prize. The new theorem addresses a different question, how votes become seats rather than how a single winner is chosen, but it carries a similar lesson: choosing an electoral system means choosing which compromises to accept.

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Notes

1. “Impossibility theorem for two-tier electoral systems.” Annals of Operations Research, 2026. DOI: 10.1007/s10479-026-07344-1.
2. University of Cambridge. “Maths shows why your vote may not count the way you think it does.” EurekAlert!, 31 July 2026. https://www.eurekalert.org/news-releases/1137946.

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