Impossibility theorem for two-tier electoral systems

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.