Mathematicians have formulated an impossibility theorem suggesting that no electoral system can perfectly satisfy three demands at once: preserving every successful local candidate’s seat, allocating parliamentary seats in exact proportion to national vote totals, and keeping the size of the legislature fixed. As political parties multiply and electoral competition becomes more fragmented, the researchers argue, at least one of these principles must eventually be weakened—unless parliaments are allowed to expand indefinitely.
The study, conducted by researchers at the University of Cambridge and the University of Copenhagen, examines both proportional representation and first-past-the-post systems. Its central finding is not that one particular voting method is uniquely flawed, but that the mathematical tension exists across electoral systems. Once the number of parties and meaningful vote distributions reaches a certain level, there may simply be insufficient seats to represent local outcomes and correct national disproportionality simultaneously.
The researchers identify three properties that are commonly associated with a fair election. The first is regionality: a candidate who wins a local contest should retain that seat. The second is proportionality: a party receiving a given share of the national vote should obtain approximately the same share of parliamentary seats. The third is a fixed parliament size, meaning that the number of legislators remains stable from one election to the next. The theorem indicates that, under increasingly fragmented conditions, all three cannot be guaranteed simultaneously.
The issue became particularly visible after Denmark’s 2022 general election. Denmark uses a two-tier proportional system in which voters elect constituency representatives and additional “levelling” seats are distributed to bring each party’s overall seat total closer to its national vote share. The system includes 40 levelling seats, but rising party fragmentation made that number insufficient to restore complete proportionality. The Social Democrats ultimately received one more seat than their vote share strictly justified, an unusual outcome in a system designed to prevent precisely that kind of mismatch.
Although a single seat may appear insignificant compared with the distortions produced by first-past-the-post elections, its political consequences can be substantial. In Denmark, the additional seat helped determine which bloc could form a government. The result demonstrated how a small mathematical discrepancy can become decisive when parliamentary majorities are narrow. According to the researchers, the problem is not evidence of manipulation or institutional failure in the conventional sense; it is a consequence of trying to fit increasingly complex vote patterns into a limited number of geographically defined seats.
Germany illustrates a different response to the same problem. Its mixed electoral system combines constituency representatives with proportional correction. When parties win more constituency seats than their national vote share would normally allow, additional seats are used to rebalance the Bundestag. After the 2021 election, the chamber expanded from its nominal size of 598 members to 736. The growth reflected the increasing number of compensatory seats required to preserve proportionality while retaining local winners. Electoral reforms adopted in 2023 sought to prevent continued expansion by placing greater emphasis on a fixed chamber size, but this also reduced the guarantee that every constituency winner would enter parliament.
The United Kingdom makes the opposite choice. Under first-past-the-post, the candidate with the most votes in each constituency wins, while national vote totals do not determine the overall allocation of seats. This can produce large differences between votes and representation, particularly when several parties compete across the same constituencies. In the 2024 general election, Labour won 411 of the 650 seats—about 63% of the House of Commons—with 33.7% of the vote. Reform UK received 14.3% of the vote but won only five seats. As more parties attract support, the researchers warn, this mismatch could become even more pronounced.
The mathematical result also helps explain why electoral reform is difficult in countries that inherited the British model. Canada has repeatedly debated replacing first-past-the-post but has retained it, while New Zealand adopted proportional representation in 1993. New Zealand’s House of Representatives does not have a permanently fixed size, allowing the system more flexibility in translating votes into seats. Similar trade-offs appear elsewhere in Europe, where proportional systems often rely on adjustment seats, flexible chamber sizes, or rules that limit the relationship between local victories and parliamentary membership.
Klausen and Holdum propose a possible alternative called “geographically ranked guaranteed proportionality.” Under the model, each party’s national seat entitlement would be determined before individual seats were assigned. A party receiving 20% of the vote, for example, would be guaranteed 20% of the seats. Constituency results would then be ranked according to the strength of each candidate’s local performance, and seats would be allocated by moving down the rankings until each party’s quota was filled. The procedure would preserve national proportionality while still using geographic results, but it would not guarantee every local winner a seat. A candidate could win comfortably in one constituency and still lose representation if the candidate’s party had already filled its national quota through stronger performances elsewhere.
The researchers say the theorem should not be interpreted as an argument that all electoral systems are equally good or equally unfair. Rather, it provides a mathematical framework for identifying what each system prioritizes and what it sacrifices. Approximate versions of regionality, proportionality, and a fixed parliament may coexist, but perfect guarantees cannot be maintained indefinitely as party systems fragment. The findings suggest that future debates over proportional representation will need to confront not only political preferences but also the underlying arithmetic of converting millions of geographically distributed votes into a finite number of legislative seats.
Subject of Research: Electoral systems, mathematical analysis, proportional representation, and parliamentary design
Article Title: Impossibility theorem for two-tier electoral systems
News Publication Date: 31 July 2026
Web References: https://doi.org/10.1007/s10479-026-07344-1
References: Annals of Operations Research, DOI: 10.1007/s10479-026-07344-1
Keywords: electoral systems, elections, proportional representation, first-past-the-post, voting mathematics, parliamentary size, regional representation, political fragmentation, mathematical analysis, political science
