A mathematical analysis of widely-used parliamentary seat apportionment methods provides detailed insights into their structural properties and possible adjustments.
Dr. Javier Cembrano, member of the department "Algorithms and Complexity". Photo: Philipp Zapf-Schramm/MPI INF
A political stakeholder should receive seats in parliament in proportion to the votes it has won—this is the basic assumption in democratic societies. However, since seats can only be assigned as whole numbers, unavoidable rounding issues arise during their distribution. This is why small changes in vote counts, or changes in the total number of seats available can have surprising effects on seat allocation. An international research team, including members from the Max Planck Institute for Informatics in Saarbrücken, Germany, has now systematically examined one of the most widely used “families” of apportionment methods, the so-called divisor methods.
The aim of the study was to gain a deeper mathematical understanding of the outcomes that can occur in seat allocation under divisor methods. In the process, the researchers discovered a surprising connection to an unsolved problem in geometry and also explored how deviations from the ideal proportional seat distribution could be minimized.
“Deciding how to allocate seats in legislative bodies has been a central topic of discussion in the political organization of democratic societies for over 200 years,” explains Javier Cembrano, a postdoctoral researcher in the “Algorithms and Complexity” department at the Max Planck Institute for Informatics. Over time, two main families of methods have emerged for distributing seats: quota methods and divisor methods.
In quota methods, the number of seats a party receives is first calculated proportionally. The seats are then assigned as whole numbers, with any remaining seats distributed to the parties with the largest remainders. However, these methods can lead to paradoxical effects, such as the “Alabama Paradox”, where a party can lose seats despite maintaining the same number of votes if the total number of seats increases.
Divisor methods avoid some of these paradoxes by dividing vote counts by a specific divisor to determine seat allocation directly. Instead of using the remainders for rounding, they employ a predefined rounding rule and look for the appropriate divisor. However, divisor methods can violate the quota principle, meaning a party might receive more or fewer seats than its rounded vote share would suggest. In Germany, a divisor method known as the Sainte-Laguë/Schepers method is used for seat allocation.
“Both method families have their strengths and weaknesses. While the space of solutions for quota methods is understood quite well, it had not been systematically explored for divisor methods,” explains Javier Cembrano. However, a deeper analysis of the solution spaces of divisor methods could improve the understanding of electoral systems and potentially lead to seat allocation procedures that more closely adhere to ideal proportionality. To this end, the researchers conducted the first systematic study of the possible seat distributions under different divisor methods.
During their investigation, they discovered a surprising connection to a long-standing open problem in geometry, more specifically in the subfield of discrete geometry, known as the “complexity of k-levels in line arrangements”. The study shows that the number of possible seat distributions under divisor methods varies in a pattern similar to the points in the k-level problem. Specifically, the researchers were able to determine upper and lower bounds for this variability and demonstrate that it remains within a defined mathematical limit. This finding reaffirms that apportionment problems are closely linked to areas of theoretical computer science and discrete mathematics. The study is therefore not only relevant for electoral systems but also discusses an algorithmic challenge, which the researchers tackled using established methods from theoretical computer science and combinatorial optimization to better understand the behavior of seat allocation methods.
Beyond the structural analysis, the researchers also examined whether randomization could help reduce deviations from the ideal proportional seat allocation. They found that randomized rounding mechanisms could, on average, lead to less distortion. However, certain deviations persist, “not to mention the fact that randomized methods would be difficult to implement in political decision-making,” notes Javier Cembrano.
The paper, titled “New Combinatorial Insights for Monotone Apportionment”, was published at the 2025 ACM-SIAM Symposium on Discrete Algorithms (SODA), one of the world’s leading conferences in theoretical computer science. Co-authors include José Correa (University of Chile), Ulrike Schmidt-Kraepelin (TU Eindhoven), Alexandros Tsigonias-Dimitriadis (European Central Bank), and Victor Verdugo (Pontifical Catholic University (PUC) Chile).
Original publication:
J. Cembrano, J. Correa, U. Schmidt-Kraepelin, A. Tsigonias-Dimitriadis, V. Verdugo (2025): „New Combinatorial Insights for Monotone Apportionment”. In: Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). https://epubs.siam.org/doi/pdf/10.1137/1.9781611978322.39
Scientific contact:
Dr. Javier Cembrano
Max Planck Institute for Informatics
Email: jcembran[at]mpi-inf.mpg.de
Editor:
Philipp Zapf-Schramm
Max Planck Institute for Informatics
Tel: +49 681 9325 5409
Email: pzs@mpi-inf.mpg.de