Mathematicians Prove the Impossibility of Completely Fair Elections 🌍✨
Article summary:
Mathematicians have reached the conclusion that a completely fair electoral system is not mathematically possible. Even the most complex methods involve trade-offs between three main elements: fair local representation, proportionality between votes and national representation, and a fixed number of parliamentary seats. This fact does not mean surrender, but rather calls for the development of balanced electoral systems that can handle these challenges more effectively.
⚖️ The dilemma of fair elections: why can’t they be fully achieved?
A joint study by the University of Cambridge and the University of Copenhagen showed that all national election systems face deep contradictions that make it impossible to achieve three important criteria at the same time:
- Regionality: ensuring that each candidate wins in their local area and secures the seat.
- Proportionality: distributing parliamentary seats in proportion to a party’s share of the vote nationwide.
- Fixed Parliament Size: maintaining a specific, fixed number of parliamentary seats regardless of changing conditions.
The researchers say that achieving all three goals together is mathematically impossible, especially with multiple political parties and increasing electoral fragmentation.
🧭 Growing pressure on electoral systems
This problem became more apparent in elections held in several Nordic countries in Europe over the past decade, where the number of political parties competing for popularity began to grow. The result was that the balance between local representation and national proportionality began to collapse, causing complex political situations.
Examples from real life:
- In Denmark 2022, the left won the election despite receiving fewer votes than the right, which sparked debate over the proportional representation system used.
- In Germany, the Bundestag saw a sharp increase in the number of seats because of the need to allocate more compensatory seats to achieve electoral proportionality, prompting the authorities to carry out reforms in 2023.
- In the United Kingdom 2024, the Labour Party won 63% of the seats despite receiving only 33.7% of the votes, the lowest proportional representation in the country’s history under the First-Past-The-Post system.
These examples reflect how electoral systems that seem fair on paper face practical challenges that make the results inaccurate and controversial.
🇩🇰 Denmark: the starting case for the study
The research project began in response to the unprecedented results in Denmark’s 2022 election, when for the first time in 75 years the votes did not match the actual distribution of seats, despite the use of a complex proportional representation system.
The system in Denmark relies on:
- Electing local representatives.
- Allocating a number of “leveling seats” to achieve proportionality between the number of seats and each party’s share of the vote at the national level.
But as the number of parties increased and supporters became more spread out, the available leveling seats at the time (only 40 seats) were not enough to adjust proportionality, which led to the Social Democrats winning one additional seat and securing the final outcome.
Dr. Frederik Ravn Klausen, one of the study’s authors, said the matter almost “broke the system,” as the two-tier system was not designed to give in to such a flaw.
🎭 Three central goals that cannot be achieved together consistently
The researchers identified three criteria that voters expect from any fair electoral system:
- Regional representation: every candidate who wins in their district should keep their seat.
- National proportionality: seats should accurately reflect the distribution of votes at the national level.
- Fixed number of seats: it is important that the number of parliamentary seats remain stable from one election to the next.
But the study showed that as the number of parties increases, the system is inevitably forced to give up one of these principles to ensure its continuity. In short: no corruption, no conspiracy, just strict mathematical calculations.
🌍 How do countries deal with these challenges?
Each country chooses to sacrifice one of these principles according to its political and social priorities.
Germany:
- Fixed seats? No.
- Its parliament expanded from 598 to 736 seats in 2021 because more compensatory seats were needed to preserve proportionality.
- The 2023 reforms stopped the parliament’s ongoing expansion but led some local winners to lose their seats.
The United Kingdom:
- It set aside national proportionality, an important aspect of its electoral system.
- Its First-Past-The-Post system makes national representation far from proportional even in major elections.
- The growth of new parties such as Reform UK helped deepen the gap between vote share and seat count.
Other countries:
- Canada retains the First-Past-The-Post system despite reform attempts.
- New Zealand adopted proportional representation in 1993 but gave up fixed seat numbers.
✨ A mathematical proposal for a fairer electoral system
The researchers proposed a new system called “Guaranteed Proportional Representation by Geographic Order”, aimed at easing some of the constraints found in traditional systems.
This system’s features include:
- Determining the number of seats each party receives based precisely on its share of the national vote.
- Distributing these seats across electoral districts ranked according to candidates’ local performance.
- Giving up full local representation, since a candidate who won locally may lose their seat if their party performs better in other districts.
This system guarantees full national proportionality but sometimes ignores the logic of regional victories; it is a realistic compromise among the three principles.
♥ Conclusion: there is no “perfect” election, but better is possible
Klausen says:
“There is no perfect electoral system, and we should not resign ourselves to that reality. Electoral systems can be designed to achieve a satisfactory balance between local representation, national proportionality, and fixed seat numbers.”
- A balanced system may not meet 100% of the criteria, but it improves electoral performance and helps reduce political controversy.
- It is important for voters and those interested in public affairs to understand these limitations of political mathematics, which directly affect election outcomes around the world.
How do we deal with the future?
With the growth of multiparty politics and changes in the political landscape, electoral systems will remain in constant renewal.
- We may see more experiments with hybrid systems that combine different principles.
- Researchers and decision-makers are focusing on finding solutions that accept carefully considered trade-offs that strengthen the legitimacy of elections.
- In the end, social dialogue and transparency remain the most important factors in ensuring voter trust.
📸 Final word
The phenomenon of the impossibility of completely fair elections is not negative in itself; rather, it opens the door to a deeper understanding of political systems and the role of mathematics in shaping society’s decisions.
Every electoral system reflects the priorities and values of the society that uses it, and we must think about how to adapt to these realities in order to develop more stable and fair democracies around the world.
Follow-up: How do these studies affect voting systems in other countries such as India and Brazil? And what is the fate of electoral systems amid rising multiparty politics?
Mathematics is only the beginning of understanding, and political reality is the challenge of implementation. 🌍🧭
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