Arithmetic exhibits that each electoral system should compromise amongst proportionality, native illustration, and parliament measurement.
Denmark’s 2022 nationwide election got here all the way down to a single seat, regardless that its proportional voting system was designed to make occasion illustration carefully match the vote. That final result factors to a deeper mathematical drawback: no electoral system can at all times protect native winners, assure proportional nationwide outcomes, and maintain parliament at a set measurement.
Researchers from the College of Cambridge and the College of Copenhagen reached that conclusion after analyzing proportional and first-past-the-post techniques. Their work exhibits that each technique of electing a nationwide legislature requires compromises, most frequently amongst regional illustration, proportionality, and the variety of parliamentary seats.
These tensions have change into more and more seen as political landscapes throughout northern Europe have fragmented into extra events. Denmark’s left bloc received the 2022 election regardless of receiving fewer votes than the precise bloc. Germany’s try to guard each native illustration and proportionality prompted the Bundestag to increase sharply earlier than reforms have been launched in 2023. Within the UK, Labour secured 63% of the seats with 33.7% of the vote in 2024, producing the nation’s least proportional basic election on document.
A wonderfully ‘honest’ system could due to this fact be not possible, however the researchers argue that electoral guidelines can nonetheless be designed to stability these competing objectives extra successfully. Their findings have been printed within the Annals of Operations Analysis.
Denmark uncovered the system’s limits
The investigation started after Denmark’s 2022 nationwide election, when votes and seats didn’t align for the primary time in 75 years. Like many European democracies, Denmark makes use of a two-tier proportional system. Voters first select native representatives, after which ‘leveling’ seats are distributed to carry every occasion’s share of parliament nearer to its nationwide vote share.
That mechanism turns into tougher to keep up as extra political events compete. Higher fragmentation requires a bigger variety of ‘leveling’ seats to protect proportionality, and the 40 accessible in Denmark’s 2022 election have been now not enough.
“Basically, it nearly broke the system,” stated research co-author Dr Frederik Ravn Klausen, from Cambridge’s Division of Pure Arithmetic and Mathematical Statistics. “The tip consequence was that the Social Democrats bought one seat greater than their votes corresponded to. In fact that’s nothing by UK requirements, however in a two-tier system, that isn’t meant to occur. It elevated the drama that this single seat determined the election.”
Klausen and co-author Sebastian Tim Holdum of the College of Copenhagen used the election as the start line for an impossibility theorem. Their consequence explains why rising occasion fragmentation is inserting established voting techniques below strain and altering actual democratic outcomes.
Three objectives can’t at all times coexist
The researchers in contrast electoral techniques used world wide, together with the UK’s first previous the publish mannequin and the proportional techniques widespread throughout Europe. They recognized three options which can be typically handled as important to a ‘honest’ election:
- Regionality: everybody who wins an area seat retains it
- Proportionality: nationwide seat totals match nationwide vote shares
- Fastened parliament measurement: the variety of seats doesn’t change from election to election
“We discovered that after there are a sure variety of political events, one in all these three parts has to offer; in any other case, the mathematics merely doesn’t work,” stated Klausen. “It’s not corruption or conspiracy, it’s merely arithmetic.”
The theory exhibits that after sufficient events are concerned, an electoral system can’t assure all three circumstances directly. Preserving two of them inevitably weakens the third.
Nations select totally different compromises
Governments have responded by deciding which precept they’re ready to calm down. Germany allowed the Bundestag to develop from 598 seats to 736 in 2021 as further leveling seats have been required to keep up proportionality.
Reforms adopted in 2023 stopped the chamber from increasing indefinitely. Nevertheless, the change eliminated the assure that each native winner would enter parliament, prioritizing a set chamber measurement as an alternative. Sweden makes use of an identical strategy.
The UK makes a unique trade-off. Its first-past-the-post system protects native winners and maintains a set variety of seats, however it doesn’t try to make nationwide seat totals proportional to nationwide vote shares.
That imbalance was much less disruptive when British politics functioned largely as a two-party system. As further events have gained assist, nonetheless, the distinction between votes and seats has widened. Reform UK acquired 14.3% of the nationwide vote in 2024 however secured solely 5 of 650 seats, whereas Labour received 411 seats with 33.7% of the vote.
“Within the subsequent British election, the votes-to-seats mismatch could possibly be even larger,” stated Klausen.
A number of former British territories inherited the identical system and have confronted comparable debates. Canada has retained first previous the publish regardless of repeated reform efforts. New Zealand adopted proportional illustration in 1993, however its Home of Representatives doesn’t have a set variety of seats.
A proportional repair sacrifices native wins
Klausen and Holdum additionally developed a doable mathematical various referred to as ‘geographically ranked assured proportionality.’ Their technique would decide every occasion’s complete variety of seats from its nationwide vote share earlier than geography is taken into account. A celebration receiving 20% of the vote would due to this fact obtain 20% of the seats.
Particular person constituencies would then be ranked in line with the power of every candidate’s native consequence. Seats could be awarded by shifting down that rating till every occasion crammed its nationwide allocation.
The strategy would assure proportionality, however it will weaken regional illustration. A candidate might win comfortably in a constituency and nonetheless lose the seat if their occasion had already reached its quota by stronger outcomes elsewhere.
“Our outcomes present that as discussions round proportional illustration within the UK get extra consideration, there isn’t a such factor as a ‘good’ system,” stated Klausen. “Nevertheless, one can nonetheless say that some techniques carry out higher than others That both proportionality, regionality, or mounted parliament measurement have to be sacrificed isn’t any excuse. One can have all three ideas roughly, but many techniques don’t.”
Reference: “Impossibility theorem for two-tier electoral techniques” by Sebastian Tim Holdum, and Frederik Ravn Klausen, 31 July 2026, Annals of Operations Analysis.
DOI: 10.1007/s10479-026-07344-1
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