By admin 
The study, published in the esteemed Annals of Operations Research, offers a stark reminder that the challenges faced by electoral systems globally are not merely political or logistical, but are rooted in fundamental mathematical constraints. As political landscapes become increasingly fragmented, with voters distributing their support across a wider array of parties, these inherent tensions are manifesting with greater frequency and severity, placing established democratic mechanisms under unprecedented strain.
The Inherent Flaws in Pursuit of Perfection
For decades, political scientists and citizens alike have grappled with the question of what constitutes a ‘fair’ electoral system. The common understanding often converges on three core ideals. First, regionality, or strong local representation, ensures that a candidate who wins a majority or plurality of votes in a specific geographic area secures a seat, providing a direct link between constituents and their representative. Second, proportionality, often considered the hallmark of a truly representative democracy, dictates that a party’s share of seats in the national legislature should closely mirror its share of the national vote. This aims to prevent situations where a minority of voters can secure a majority of power. Third, a fixed parliament size is crucial for stability, efficiency, and public trust, avoiding the logistical and political complications of an ever-expanding or shrinking legislative body.
However, the impossibility theorem articulated by Dr. Frederik Ravn Klausen from Cambridge’s Department of Pure Mathematics and Mathematical Statistics, and Sebastian Tim Holdum of the University of Copenhagen, demonstrates that once a political system includes a certain number of parties, these three goals cannot simultaneously and perfectly coexist. "We found that once there are a certain number of political parties, one of these three components has to give, otherwise the maths simply don’t work," explained Dr. Klausen. "It’s not corruption or conspiracy, it’s simply mathematics." This revelation echoes the spirit of other famous impossibility theorems in social choice theory, such as Arrow’s Impossibility Theorem for voting systems, which demonstrate fundamental limits to achieving ideal collective outcomes.
Global Democracies Under Growing Pressure
The strains on electoral systems have become increasingly evident in recent years, particularly in northern European democracies known for their sophisticated proportional representation models. The rise of multi-party politics, often fueled by shifting ideologies, identity politics, and increased voter volatility, means that traditional two-party or even three-party dominance is giving way to more diverse political landscapes. This fragmentation means that vote shares are spread thinner across more contenders, making the task of translating votes into seats while upholding all three principles exponentially harder.
The Danish Anomaly: A System Stretched to its Limit
The inspiration for this pivotal research stemmed directly from Denmark’s 2022 national election, an event that sent ripples through the country’s political establishment. For the first time in 75 years, Denmark’s highly regarded two-tier proportional representation system failed to produce a fully proportional outcome. Like many European nations, Denmark employs a system where voters first elect local representatives in constituencies. To ensure national proportionality, an additional layer of ‘leveling’ seats is then distributed among parties to adjust their parliamentary representation closer to their overall national vote share.
In 2022, despite receiving fewer votes nationally, the left bloc emerged victorious over the right bloc. This occurred because the 40 available leveling seats were insufficient to fully correct the disproportionality arising from the distribution of constituency seats and the fragmented vote. "Essentially, it almost broke the system," remarked Dr. Klausen. "The end result was that the Social Democrats got one seat more than their votes corresponded to. Of course that is nothing by UK standards, but in a two-tier system, that isn’t meant to happen. It increased the drama that this single seat decided the election." This incident underscored how even highly refined proportional systems, designed specifically to ensure fairness, can buckle under the pressure of increasing political diversity, directly impacting governmental outcomes and public trust.
Germany’s Expanding Bundestag: The Fixed Size Dilemma
Germany’s experience with its mixed-member proportional (MMP) system offers another compelling case study of these mathematical constraints in action. The German system allows voters two votes: one for a direct constituency candidate and another for a party list. This structure aims to combine the benefits of local representation with national proportionality. However, the mechanism designed to achieve this — particularly the allocation of "overhang" and "leveling" seats — led to an unprecedented expansion of the Bundestag.
By 2021, the Bundestag had swelled to 736 seats, significantly larger than its intended 598. This happened because as more parties gained support and more local winners emerged from parties that might not have otherwise earned many seats through the party list vote, an increasing number of "leveling" seats were required to maintain the desired proportionality. The political and financial costs of an ever-growing parliament became unsustainable, leading to extensive debates and, ultimately, significant reforms in 2023. These reforms, while preventing indefinite expansion, controversially removed the absolute guarantee that every local winner would enter parliament, thereby sacrificing a degree of regionality to preserve a fixed parliament size and proportionality. Sweden employs a similar approach, indicating a broader trend among countries grappling with this particular trade-off.
The UK’s First-Past-the-Post: A Proportionality Sacrifice
In stark contrast to continental Europe, the United Kingdom’s first-past-the-post (FPTP) system epitomizes the sacrifice of proportionality. Under FPTP, the candidate with the most votes in each constituency wins, regardless of their overall percentage of the vote. For much of the 20th century, when British politics largely operated as a two-party system, FPTP was often defended for its simplicity and its tendency to produce strong majority governments. However, as the political landscape diversified, with the emergence of significant third parties and a more diffuse distribution of voter support, the disjuncture between votes and seats has become increasingly pronounced.
The 2024 UK general election provided the most dramatic illustration of this disproportionality on record. The Labour Party secured 63% of the seats in the House of Commons with just 33.7% of the national vote. Conversely, Reform UK, a rapidly growing party, garnered 14.3% of the national vote but secured only five of the 650 seats. This stark imbalance means millions of votes are effectively "wasted," leading to widespread voter disillusionment and persistent calls for electoral reform from various advocacy groups like "Make Votes Matter." The mathematical findings predict that "in the next British election, the votes-to-seats mismatch could be even bigger," according to Dr. Klausen, suggesting that the UK’s system is becoming increasingly ill-suited to its multi-party reality.
Several former British territories inherited the FPTP model and have similarly faced pressure for change. While Canada has retained FPTP despite repeated reform efforts, New Zealand, for instance, adopted a form of proportional representation (MMP) in 1993, implicitly acknowledging the limitations of FPTP. However, New Zealand’s system does not guarantee a fixed parliament size, further illustrating the pervasive nature of these three conflicting goals.
A Mathematical Proposal for Fairer Elections
Acknowledging the inherent limitations of existing systems, Klausen and Holdum have also proposed a potential alternative, an algorithm called ‘geographically ranked guaranteed proportionality’. This innovative approach is designed to circumvent some of the fundamental trade-offs embedded in current electoral models.
Under this proposed system, each party’s national seat total would be determined entirely by its overall national vote share. For example, a party receiving 20% of the national vote would precisely receive 20% of the parliamentary seats, irrespective of where those votes were cast geographically. To then allocate these seats, individual constituencies would be ranked according to how strongly each candidate performed locally. Seats would then be awarded by moving down this ranking until each party had filled its national allocation.
This method would rigorously guarantee proportionality, fulfilling one of the three desired goals without compromise. However, it would necessarily weaken regionality. A popular candidate could win comfortably in their local constituency, yet still fail to enter parliament if their party had already filled its national quota with stronger performances from candidates in other regions. This highlights that even in a mathematically optimized system, a choice must be made regarding which principle to prioritize or, conversely, to partially sacrifice.
No Perfect System, But Better Ones Are Possible
The findings of Klausen and Holdum underscore a critical lesson for democracies worldwide: the pursuit of a ‘perfect’ electoral system, one that simultaneously achieves absolute regionality, proportionality, and a fixed parliament size, is a mathematical impossibility. "Our results show that as discussions around proportional representation in the UK get more attention, there is no such thing as a ‘perfect’ system," Dr. Klausen reiterated.
However, the research is not an endorsement of electoral fatalism. Instead, it offers a pragmatic framework for reform. While a perfect system may be unattainable, the study strongly argues that "one can still say that some systems perform better than others. That either proportionality, regionality, or fixed parliament size must be sacrificed is no excuse. One can have all three principles approximately, yet many systems do not."
The implications are profound. Understanding these mathematical limits empowers countries to design electoral systems that more effectively balance the three competing goals, even if absolute perfection remains elusive. It shifts the conversation from an idealistic quest for a flawless system to a realistic and informed debate about which compromises are most acceptable and beneficial for a given society. As political fragmentation continues to challenge democratic norms, this mathematical insight provides a crucial foundation for strengthening electoral integrity and fostering greater trust in democratic institutions. Dr. Frederik Ravn Klausen is a Member of Clare Hall, Cambridge.
