Largest remainder method

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 (i.e. usually the total number of votes cast divided by the number of seats, or some similar formula). The result for each party will usually consist of an integer part plus a fractional remainder. Each party is first allocated a number of seats equal to their integer. This will generally leave some remainder 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.

There are several possibilities for the quota. The most common are: the Hare quota and the Droop quota. The use of a particular quota with the largest remainders method is often abbreviated as "LR-[quota name]", such as "LR-Droop".[2]

The Hare (or simple) quota is defined as follows

${\displaystyle {\frac {\text{total votes}}{\text{total seats}}}}$

It is used for legislative elections in Russia (with a 5% exclusion threshold since 2016), Ukraine (5% threshold), Bulgaria (4% threshold), Lithuania (5% threshold for party and 7% threshold for coalition), Tunisia,[3] Taiwan (5% threshold), 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.[4] 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

${\displaystyle 1+{\frac {\text{total votes}}{1+{\text{total seats}}}}}$

and is applied in elections in South Africa. The Hagenbach-Bischoff quota is virtually identical, being

${\displaystyle {\frac {\text{total votes}}{1+{\text{total seats}}}}}$

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.[5][6][7][8][9] However, an example shows that the Hare quota can fail to guarantee that a party with a majority of votes will earn at least half of the seats (though even the Droop quota can very rarely do so).

The Imperiali quota

${\displaystyle {\frac {\text{total votes}}{2+{\text{total seats}}}}}$

is rarely used since it suffers from the defect that it might result in more seats being allocated than there are available (this can also occur with the Hagenbach-Bischoff quota but it is very unlikely, and it is impossible with the Hare and Droop quotas). 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.

These examples take an election to allocate 10 seats where there are 100,000 votes.

The largest remainder method satisfies the quota rule (each party's seats amount to its ideal share of seats, either rounded up or rounded down) 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 (with the number of votes held constant), parties D and E counterintuitively end up with fewer seats.

With 25 seats, the results are:

With 26 seats, the results are:

• Hamilton method experimentation applet at cut-the-knot