It has been found that a 'perfectly fair election' is mathematically impossible, as it cannot simultaneously satisfy the three conditions of regional representation, proportional representation of votes, and the number of seats.
It has been proven mathematically impossible to create an electoral system that consistently guarantees three seemingly fair conditions in all election results: 'the candidate with the most votes locally wins,' 'seats are allocated to each party according to their national vote share,' and 'the number of seats in parliament remains unchanged.' According to a research team from Cambridge University and the University of Copenhagen, if the number of political parties is sufficiently large compared to the number of supplementary seats, it will be necessary to give up one of the following: 'regional representation,' 'seat allocation proportional to votes,' or 'a fixed number of seats.'
Impossibility theorem for two-tier electoral systems | Annals of Operations Research | Springer Nature Link
Mathematicians prove perfectly fair elections are impossible | ScienceDaily
https://www.sciencedaily.com/releases/2026/08/260801042812.htm
Suppose one party wins in many constituencies by a narrow margin of 51% to 49%, while another party wins overwhelmingly in a few constituencies by securing 80% of the vote. In single-member constituencies, only the candidate with the most votes wins a seat regardless of the margin of victory. Therefore, a party that achieves close victories across a wide area may end up with a higher percentage of seats than its national vote share would suggest. By dividing constituencies in a way that favors a particular party, it is possible for a party with a low national vote share to win a majority of seats.
'Gerrymandle' is a puzzle game where players draw electoral district boundaries to win elections by securing more seats than their opponents - GIGAZINE
When allocating seats to political parties based on their nationwide vote share, it's necessary to give additional seats to parties that are short on seats. However, if the number of seats is fixed, there's a limit to how many can be added. The impossibility theorem presented by the research team shows that if there are multiple electoral districts and the number of political parties is sufficiently large compared to the number of supplementary seats, at least one of the three conditions will no longer be met.
While it's intuitively easy to imagine situations where the three conditions conflict, the research team's achievement lies in mathematically proving that this problem is not limited to specific countries or electoral systems. By clearly defining regional representation, proportionality, and a fixed number of seats, and demonstrating that no system exists that guarantees all three conditions in every election outcome, even with careful allocation methods, they revealed the inherent limitations of electoral systems themselves, rather than flaws in the design of individual systems.
In a real-world example, in Denmark's 2022 general election, 40 supplementary seats were not enough to close the gap, resulting in the Social Democratic Party winning one more seat than it should have received. Although the difference in the number of seats was only one, it led to the left-wing camp gaining a parliamentary majority with fewer votes than the right-wing camp. In Germany, the number of seats has continued to increase in order to balance regional winners with proportional representation, reaching 736 seats in 2021, far exceeding the benchmark of 598 seats.
The research team proposes 'Geographically Ranked Guaranteed Proportionality' as a method to prioritize proportionality. This system first determines the total number of seats each party will have based on the total number of votes received nationwide, then ranks each party's vote performance in each region, and allocates seats starting with the top-ranked regions. This ensures a proportional distribution of seats nationwide while allocating seats as closely as possible to the election results in each region.
However, if all seats allocated to a political party are filled, even the candidate with the most votes in a constituency may lose. This system protects proportionality and the number of seats, but it does not guarantee that a victory in a local area will necessarily lead to a seat in parliament.
The research team stated that 'it is impossible to guarantee regional representation, proportionality, and a fixed number of seats,' but argued that 'a system that largely satisfies the three conditions can be realized.' They said that what is required of an electoral system is not complete fairness, but rather a clear understanding of what to prioritize and what inequalities to accept.
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