The largest remainder method requires the numbers of votes for each party to be divided by a quota representing the number of votes required for a seat. The result for each party will usually consist of an integer part plus a fractional remainder. Each party is first allocated a number of seatsequal to their integer. This will generally leave some seats unallocated: the parties are then ranked on the basis of the fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all the seats have been allocated. This gives the method its name.
Quotas
There are several possibilities for the quota. The most common are: the Hare quota and the Droop quota. The Hare quota is defined as follows It is used for legislative elections in Russia, Ukraine, Tunisia, Taiwan, Namibia and Hong Kong. The Hamilton method of apportionment is actually a largest-remainder method which uses the Hare Quota. It is named after Alexander Hamilton, who invented the largest-remainder method in 1792. It was first adopted to apportion the U.S. House of Representatives every ten years between 1852 and 1900. The Droop quota is the integer part of and is applied in elections in South Africa. The Hagenbach-Bischoff quota is virtually identical, being either used as a fraction or rounded up. The Hare quota tends to be slightly more generous to less popular parties and the Droop quota to more popular parties. This means that Hare can arguably be considered more proportional than Droop quota. The Imperiali quota is rarely used since it suffers from the defect that it might result in more seats being allocated than there are available. This will certainly happen if there are only two parties. In such a case, it is usual to increase the quota until the number of candidates elected is equal to the number of seats available, in effect changing the voting system to the Jefferson apportionment formula.
Examples
These examples take an election to allocate 10 seats where there are 100,000 votes.
Hare quota
Droop quota
Pros and cons
It is relatively easy for a voter to understand how the largest remainder method allocates seats. The Hare quota gives an advantage to smaller parties while the Droop quota favours larger parties. However, whether a list gets an extra seat or not may well depend on how the remaining votes are distributed among other parties: it is quite possible for a party to make a slight percentage gain yet lose a seat if the votes for other parties also change. A related feature is that increasing the number of seats may cause a party to lose a seat. The highest averages methods avoid this latter paradox; but since no apportionment method is entirely free from paradox, they introduce others like quota violation.
Technical evaluation and paradoxes
The largest remainder method satisfies the quota rule and was designed to satisfy that criterion. However, that comes at the cost of paradoxical behaviour. The Alabama paradox is exhibited when an increase in seats apportioned leads to a decrease in the number of seats allocated to a certain party. In the example below, when the number of seats to be allocated is increased from 25 to 26, parties D and E counterintuitively end up with fewer seats. With 25 seats, the results are:
