Trade offs of two-tier electoral systems from a mathematical perspective

Recommendable! The larger the parliament, the better the representation. However, what size of parliament to choose?

"The researchers ... 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 [after unification], 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. ...

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

..."

From the abstract:

"Two-tier electoral systems aim to balance regional representation and proportionality. Recently, the mathematical mechanisms of several Northern European two-tier systems have been challenged by increasing political party fragmentation, as evidenced by the expanding German parliament and the majority-deciding disproportionality of Denmark’s 2022 election. Regionality, proportionality, and two-tier systems are defined mathematically and used to identify the common cause behind these developments through the formulation of an impossibility theorem: no electoral system with fixed parliament size can guarantee both proportional and geographic representation.

Finally, a new algorithm, geographically ranked guaranteed proportionality, is proposed to circumvent the impossibility theorem."

Maths shows why your vote may not count the way you think it does | University of Cambridge "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."

Impossibility theorem for two-tier electoral systems (open access)

Fig. 5 From: Impossibility theorem for two-tier electoral systems

Other systems include two-round and run-off systems (USA , France etc.), preferential voting, transferable votes, and non-democracies without elections.