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<p>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, <i>geographically ranked guaranteed proportionality</i>, is proposed to circumvent the impossibility theorem.</p>

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Impossibility theorem for two-tier electoral systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月31日

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  • Sebastian Tim Holdum,
  • Frederik Ravn Klausen

摘要

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.