When we say an election is fair, we are usually bundling several different things together. We want whoever wins a seat in their own constituency to keep it. We want the overall number of seats held by each party to reflect the votes it won across the country. And we want Parliament to have a fixed size, without adding places every time the numbers don't add up.
Three reasonable demands. According to a theorem formulated by Sebastian Tim Holdum of the University of Copenhagen and Frederik Ravn Klausen of the University of Cambridge, they cannot all be guaranteed at once. When parties become numerous enough relative to the available seats to correct imbalances, at least one of the three conditions must give way.
The title of this article requires a qualification. Mathematics has not shown that every conceivable idea of electoral fairness is unachievable. It has identified a precise incompatibility in systems that try to combine territorial representation with national proportionality while keeping the number of parliamentarians fixed.
One example is enough to show the problem. Suppose a party, based on its national share of the vote, is entitled to ten seats. In the constituencies it has already won eleven. If all the locally elected members keep their seats, that party will be overrepresented. To restore proportionality you can strip a seat from one of the local winners, give the other parties fewer seats than their share entitles them to, or add a deputy. Either way, one of the original promises is broken.
Two-tier systems try to sidestep this conflict. They first allocate a portion of seats at the constituency level, then distribute compensatory seats to parties that fell short of their national quota. As long as the imbalances are modest, the correction works. But the available seats are not infinite. With more parties and more fragmented local results, a point may come where they are simply not enough.
That is what happened in Denmark in 2022. For the first time since 1947, a seat won at the regional level could not be compensated by the remaining available places. The Social Democrats ended up with one deputy more than a strictly proportional allocation would have given them. That single seat decided the parliamentary majority: it was no longer a small imperfection buried in the arithmetic, but a politically decisive outcome.
Denmark is not some quirky exception. Before the 2023 reform, Germany protected both local winners and proportionality by adding seats to the Bundestag. The standard size was 598, but after the 2021 election the deputies numbered 736. The solution honoured two principles by letting the third collapse: a fixed parliamentary size.
The German reform chose the opposite compromise. It capped the size of the assembly but no longer guarantees that every constituency winner automatically enters Parliament. Sweden too amended its rules after compensatory seats proved insufficient. Many other systems, by contrast, maintain a fixed size and local representation while accepting less proportional national outcomes.
This does not make all electoral laws equivalent. Some approximate the three objectives well; others produce far greater distortions. The authors themselves propose a method for reducing the conflict. The theorem does not provide cover for whoever designs a system badly; it only prevents anyone from promising that a single formula can always preserve everything.
Politics tends to frame electoral reform as the search for a neutral mechanism: just find the right threshold, the correct number of constituencies, or the best algorithm. Mathematics shows that part of the conflict comes before the formulas. Territory, proportionality and assembly stability are different ideas of representation. When they clash, the law must decide which one counts most.
Mathematics does not say which system we should choose. It only says that, before choosing, we should openly declare which idea of fairness we are prepared to give up.
