Ensuring Fair Elections With the Help of Mathematics


RWTH graduate receives the KlarText Award for Science Communication for his doctoral thesis on optimizing the design of constituencies for the Bundestag elections.


Dr. Sebastian Goderbauer completed his doctorate at the RWTH Chair of Operations Research headed by Professor Marco Lübbecke. His dissertation is entitled "Mathematical Optimization for Optimal Decision-Making in Practice: Energy Systems and Political Districting." Among other things, the thesis is concerned with the delimitation of electoral districts for the German Bundestag elections using a mathematical procedure.

Now Goderbauer has received the KlarText Prize for Science Communication in the field of mathematics, worth 7,500 euros. The Klaus Tschira Foundation awards the prize annually to early-career researchers from the fields of biology, chemistry, computer science, geosciences, mathematics, neurosciences, and physics who succeed in explaining their doctoral research to a lay audience in an easily understandable way.

A Problem: The Size of the Bundestag

The doctoral topic also addressed a current problem of German politics: the size of the Bundestag, the German parliament. Due to the increase in overhang and leveling seats, the parliament is growing in size with every election and now comprises 709 members, far more than the statutory number of 598. A solution is offered by a reform of electoral law, which may also include a reduction in the number of constituencies and thus a redesign of the “electoral map,” so to speak. Federal Election Commissioner Dr. Georg Thiel makes use of the results of Goderbauer's work.

Currently, the Bundestag is elected in 299 electoral districts, which are tailored by the legislators before each election. The districting requirements are regulated by electoral law: Constituencies should consist of geographically contiguous areas, be based on existing administrative boundaries, and be as equal in population as possible. In practice, the electoral districts are still “manually” defined, with less attention paid to equality in population.

Goderbauer’s dissertation shows that the application of a mathematical optimization method offers distinct advantages over such ‘manual’ districting. In the context of the current reform debate, various constituency scenarios can be quickly simulated.

“Commissioned by the Federal Returning Officer, we have created maps with 270, 250, 200, and 125 optimized electoral districts, based on our research and with the aim of fully meeting the legal requirements,” said Goderbauer. “Politicians, however, tend to be critical of our results, because a reform of the electoral district boundaries may affect their chances of re-election.” For example, a modified population structure could have an impact on election results, if electoral districts are more rural or urban than before, for example.

However, a mathematically calculated definition of electoral districts based on the legal requirements alone provides an objective and fair solution. The fact that the current electoral districts for the Bundestag elections do not fully meet the legal partitioning requirements is not an instance of deliberate manipulation, but results from the complexity of the problem. It is just not possible to solve this task manually in an optimal way, explains Goderbauer: “We therefore wanted to examine the current definition of districts from a mathematical perspective. From a scientific point of view, the partitioning process corresponds to a so-called optimization problem.”

In order to solve this problem, the researchers translated the legal regulations into the language of mathematics. “From a purely mathematical point of view, this political districting problem can be interpreted as a decomposition task for an abstract graph-theoretical structure. In this context, a graph represents a set of objects and existing connections between them. We developed the graph based on a map of Germany.” The researchers marked all the cities with a dot in their center and connected neighboring points with a line. The resulting graph then had to be broken down into, say, 299 parts, which , on the basis of the map, were interpreted as electoral districts.

However, there are more decision possibilities for this partitioning task than there are atoms in the universe, and this is not unusual for a complex optimization task with specific constraints. The mathematicians therefore used an algorithm that searches the solution space section by section. "Our research shows that all legal requirements can be met very well in this way and that a fair division of constituencies is mathematically possible," said Goderbauer.