There Can Be No Perfect Electoral System, Mathematicians Demonstrate

The Democratic Dilemma: Why a Perfect Voting System Is Mathematically Impossible

According to НВ — Техно: Published: August 3, 10:00 PM

A study by mathematicians at the University of Cambridge and the University of Copenhagen, published in the journal Annals of Operations Research, shows that a completely fair electoral system cannot exist. The authors formulate an impossibility theorem: no voting method can simultaneously guarantee that the winner of a local race keeps their seat, that a party's national share of the vote is accurately reflected in parliament, and that the total number of lawmakers stays fixed. This is a timely reminder for democracies debating election reform: every system involves unavoidable trade-offs.

Looking at Real-World Voting Systems

Researchers Frederik Ravn Klausen and Sebastian Tim Holdum examined proportional and majoritarian election models around the world. Their work demonstrates that the clash between regional representation, national proportionality, and the size of a legislature comes from the laws of mathematics, not from corruption or conspiracy. Examples include:

• In Denmark's 2022 election, the left-wing bloc won a majority of seats despite receiving fewer votes overall than its opponents.
• In Germany, attempts to preserve balance caused the Bundestag to expand from 598 to 736 seats in 2021.
• In the United Kingdom's 2024 election, Labour captured 411 of 650 parliamentary seats, or 63%, with only 33.7% of the vote. Reform UK, in contrast, gained 14.3% of the vote but just 5 seats.

Based on these findings, the researchers propose an algorithm called geographically ranked guaranteed proportionality. It ensures that a party's number of mandates matches its national vote share, but at the cost of weakening the connection between representatives and particular regions. The overall conclusion: no flawless electoral system exists. Each model has its own advantages and disadvantages, and further study and improvement are needed.

The research calls into question traditional assumptions about the effectiveness of electoral systems and emphasizes the need to adapt and refine existing models for greater fairness. In light of these results, political parties and legislators may be forced to rethink their approach to elections so that voters' intentions are better represented. In the future, this could influence the political systems of many countries seeking more transparent and just elections.