So far we’ve seen what happens when we do try to apply the majority vote to establish a global preference hierarchy starting from the individual voter preferences. This does not mean that applying a purely proportional criterion saves us from paradoxes and mathematical curiosities. Generally the problems are related to the fact that applying a proportional system must in some way makes partitions between the available seats and those entitled to vote to create the proportion.
In doing so, divisions will produce remainders and the method chosen for distribute those remainders of the divisions can create unexpected results. Everything is complicated by the fact that each electoral college has to receive at least one seat, albeit much smaller than the most populous seats. Let’s see some historical examples that have emerged in the recent US history but are conceptually applicable to any proportional distribution system.

This is a paradox discovered in the United States during the 1880 census by C.W. Seatonchief, clerk of the United States Census Bureau. As the US constitution quotes: “The number of Representatives should not exceed one in thirty thousand people, but each state must have at least one representative”, over the years, the number of parliamentary representatives and the law for the apportionment of the seats has often been revised to adapt to the dictates of the constitution, or to some of the political parties. In 1880 was still in force the Hamilton’s method that was first adopted in 1852. This method provided that the number of seats was allocated proportionally by firstly dividing the total number of voters by the total number of seats and then multiplying the divisor thus obtained for the population of the individual state. Since the division does not tend to give a whole result, there remains unassigned seats that are redistributed among the states still proportionally, giving the largest number of seats to the state with the highest remainder and so on. Seaton calculated the number of representatives each state would send to the parliament, assuming a total number of parliamentarians from 275 to 350. He found during the calculation that a parliament of 299 representatives would have provided Alabama with 8 seats, but increasing by one the number of representatives to 300 Alabama would lose a seat instead of keeping unchanged or increasing its quota. Let’s give it an example.

State Population Proportion [%] Seats
A 6 42.86 4
B 6 42.86 4
C 2 14.29 2
tot 14 10

Suppose we now increase the number of available seats. We expect a better distribution of seats with the result of favoring the state with less representativity…

Stato Popolazione Proporzione [%] Seggi
A 6 47.14 5
B 6 47.14 5
C 2 15.71 1
tot 14 11

…instead no! The state with less representation is penalized by losing a seat. This is because as the number of seats grows, the proportional share grows faster for larger states than the smaller ones.

This is a case very similar to the previous one, only in this case, rather than suppose to increase the number of seats in parliament, let’s assume that the population of some states increases. Let’s see what can happen.

State Population Seats Population Seats
A 92 15.44→15 97 15.75→16
B 42 7.05→7 41 6.66→7
C 15 2.52→3 16 2.60→2
tot 149 25 154 25

In this case we have very counterintuitive results. In fact, state C grew by 6.6% but lost a seat while state A, which grew by only 5.5%, earned one. State B instead, while losing citizens, has maintained its share of representation. This paradox was discovered around the 1900 in the United States when Virginia, though growing much Faster than Maine (about 60% faster), lost a seat while the Maine earned one.