DEMOCRACY WATCHMath Shows Why Your Vote May Not Count the Way You Think It Does

Mathematicians have shown that completely fair elections are – in a sense – impossible. No electoral system can guarantee both that local votes translate directly into local seats, and that total seat distribution matches overall vote shares: not without an indefinitely growing parliament. 

The researchers, from the University of Cambridge and the University of Copenhagen, analysed different electoral systems, both proportional and first past the post, and found that in any system, there are always going to be a series of trade-offs when electing a national legislature. For most systems, these trade-offs are between regional representation, guaranteed proportionality and parliamentary size. 

In the past decade, voters in northern European democracies have seen the results of these trade-offs, driven by the increasing fragmentation of political parties. In Denmark, the left bloc won the 2022 election despite winning fewer votes than the right bloc, even though Denmark has a proportional system. In Germany, the desire to balance regional representation and proportionality led to the Bundestag ballooning in size, before reforms were made in 2023. And in the UK, the 2024 general election was its least proportional ever, as Labour won 63% of the seats with 33.7% of the vote.

However, while there may not be a completely ‘fair’ electoral system, the researchers say there are ways to get around the tension between regionality, proportionality and parliament size, and make the impossible, possible. Their results are reported in the Annals of Operations Research. 

The researchers began their study after the 2022 Danish national election, where the number of votes did not correspond to the number of seats for the first time in 75 years. The Danish system, like many in Europe, is a two-tier or proportional system. Voters elect their local representatives, and then a number of ‘levelling’ seats are added so that each party’s seat share roughly matches its vote share. But as party fragmentation increases, the number of ‘levelling’ seats also needs to increase to maintain that proportionality. In the 2022 Danish election, the 40 levelling seats were no longer enough to achieve proportionality.