Math Shows Why Your Vote May Not Count the Way You Think It Does
“Essentially, it almost broke the system,” said study co-author Dr Frederik Ravn Klausen, from Cambridge’s Department of Pure Mathematics and Mathematical Statistics. “The end result was that the Social Democrats got one seat more than their votes corresponded to. Of course that is nothing by UK standards, but in a two-tier system, that isn’t meant to happen. It increased the drama that this single seat decided the election.”
Working with his co-author Sebastian Tim Holdum from the University of Copenhagen, Klausen formulated an impossibility theorem that illustrates how different voting systems are straining under increased fragmentation, which is having real effects on democratic outcomes.
The researchers looked at different electoral systems around the world, both first-past-the-post like the UK, and proportional systems like much of the rest of Europe. They identified three main components of a ‘fair’ system:
● Regionality: everyone who wins a local seat keeps it
● Proportionality: national seat totals match national vote shares
● Fixed parliament size: the number of seats doesn’t change from election to election
“We found that once there are a certain number of political parties, one of these three components has to give, otherwise the maths simply don’t work,” said Klausen. “It’s not corruption or conspiracy, it’s simply mathematics.”
Different countries have made different choices when it comes to which of the three principles they’re willing to sacrifice. In Germany, the Bundestag grew from 598 seats to 736 seats in 2021, because the required number of levelling seats kept growing. In 2023, the German system was reformed to prevent an ever-expanding Bundestag, but it sacrificed the guarantee of a local representative to keep the chamber a fixed size. Sweden is among the countries that has adopted a similar model.
In the UK, which has a first-past-the-post system, proportionality is the component which gets sacrificed. For most of the last century, the UK has been in essence, a two-party system, but as new parties form and grow across the political spectrum, it leads to ‘unfair’ results. In 2024, Reform UK won 14.3% of the vote but won just five of 650 seats, while Labour won 411 seats on 33.7% of the vote. “In the next British election, the votes-to-seats mismatch could be even bigger,” said Klausen.
Many former British territories inherited the British system and are also struggling to adapt. Canada has kept the first-past-the-post system through repeated attempts at reform. New Zealand switched to proportional representation in 1993 – but the New Zealand House of Representatives does not have a fixed parliament size.
However, Klausen says there could be a potential solution, also backed up by maths. He and Holdum have proposed an algorithm they call ‘geographically ranked guaranteed proportionality’ which could get around the impossibility of current systems. In their proposal, national totals are set before any voting takes place and irrespective of geography, so that any party that gets 20% of the vote gets 20% of the seats. Next, every individual seat is ranked by strength of local performance, and seats are allocated by working down that list.
However, this system, while proportional, does sacrifice regionality, because a candidate could win their seat comfortably, but if their party has already used up their quota on stronger wins elsewhere, that candidate would not get to keep their seat.
“Our results show that as discussions around proportional representation in the UK get more attention, there is no such thing as a ‘perfect’ system,” said Klausen. “However, one can still say that some systems perform better than others. That either proportionality, regionality and fixed parliament size must be sacrificed is no excuse. One can have all three principles approximately, yet many systems do not.”
