来自剑桥大学和哥本哈根大学的数学家通过研究证明，在数学上不存在能够同时满足所有公平性要求的选举制度[1]。研究表明，当政党数量足够多时，任何投票方法都无法兼顾地方代表制、全国比例代表制和固定议会规模这三个目标[1]。该研究已发表在《运营研究年刊》2026年期刊上[1]。
研究人员以多个国家的真实案例论证了这一不可能性[1]。在丹麦2022年选举中，社会民主党获得的议席比其投票份额对应的席位多一席，这是75年来首次出现[1]；英国2024年大选中，改革英国党虽获得14.3%的全国投票率，但仅在650个议席中获得5个席位，而工党以33.7%的投票率赢得411个议席[1]；德国联邦议院则因无法协调这些冲突目标，于2021年将议席从598个扩至736个[1]。
鉴于完全公平的制度本质上不可行，研究团队提出了一种新的投票算法——"地理排名保证比例代表制"，用以在这些不可避免的权衡中寻求最优平衡[1]。
Researchers from Cambridge University and the University of Copenhagen have demonstrated mathematically that no voting system can simultaneously achieve three fundamental democratic principles [1]. When the number of political parties is sufficiently large, any electoral method must sacrifice at least one of the following objectives: maintaining local representation where winners retain constituency seats, ensuring national proportionality between total votes cast and parliament seats allocated, or keeping the size of parliament fixed [1].
The findings, published in the Annals of Operations Research in 2026, reveal the inherent trade-offs embedded in electoral design [1]. Real-world examples illustrate these compromises: in Denmark's 2022 election, the Social Democratic Party secured one more seat than its national vote share would mathematically warrant, marking the first such occurrence in 75 years [1]. The 2024 UK general election demonstrated similar disproportionality, with Reform UK earning 14.3 percent of the national vote but only 5 of 650 parliamentary seats, while Labour obtained 33.7 percent of votes yet won 411 seats [1]. Germany's response has been structural; the Bundestag expanded from 598 seats in 2021 to 736 seats to better accommodate these competing demands [1].
The research team proposed a new voting algorithm called "geographically ranked guaranteed proportionality" to help policymakers navigate these unavoidable trade-offs [1]. The work underscores that electoral systems cannot be perfected but rather must be consciously designed to balance competing fairness criteria according to democratic priorities.