Historical article: This text is based on an older article from Klein-Singen.de. It documents the debates and data available at the time and has not been silently rewritten as if it described the present.
A proportional electoral system must distribute a limited number of parliamentary seats according to parties’ shares of the vote. The task would be trivial if the number of seats equalled the number of voters. Because a parliament contains far fewer seats than votes, fractional entitlements must somehow be converted into whole numbers.
That conversion creates the possibility of paradoxes. Different methods use different rounding principles. Mathematically, one distinguishes broadly between quota methods and divisor methods.
- D’Hondt: a divisor method associated with systematic downward rounding in its quotient representation.
- Sainte-Laguë: a divisor method using standard rounding.
- Hare/Niemeyer: a largest-remainder quota method.
The Hare/Niemeyer method
For each party, calculate the quota
party votes / total valid votes × total number of seats
The integer part is allocated first. Remaining seats are then awarded according to the largest fractional remainders.
This looks intuitively fair, but largest-remainder methods can exhibit surprising effects.
The Alabama paradox
If the total number of seats in a parliament is increased while the vote distribution remains unchanged, a party can lose a seat. This became famous through an historical apportionment problem in the United States and is therefore known as the Alabama paradox.
The original German article illustrated the effect with Bundestag election figures: when the total number of seats changed from 656 to 657, one party could lose a seat even though no party’s vote total had changed. The additional seat changed the ordering of the fractional remainders.
The new-states paradox
Adding or removing a party together with its votes and corresponding seats can alter the seat allocation among the remaining parties. In electoral systems with thresholds, a party entering or leaving the allocation pool can therefore affect competitors in ways that are not locally intuitive.
The population paradox
A party can gain votes relative to another party and nevertheless lose a seat to it under some apportionment procedures. More generally, changing the vote total of one party can alter the allocation between other parties whose own totals remain unchanged.
Why these effects occur
The paradoxes are not arithmetic mistakes. They result from the interaction of:
- fractional proportional entitlements;
- a fixed integer number of seats;
- a specific rounding or remainder rule.
No apportionment method can satisfy every attractive fairness criterion simultaneously. The choice of method is therefore not merely technical; it determines which mathematical properties are guaranteed and which paradoxes remain possible.
The historical article used Hare/Niemeyer because it was directly relevant to German electoral practice at the time. Today, the broader mathematical lesson is the more durable one: rounding rules can have political consequences.