Largest remainders 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. Usually, this is given by the total number of votes cast, divided by the number of seats. 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. To apportion these seats, the parties are then ranked on the basis of their fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all seats have been allocated. This gives the method its name.

There are several possible choices for the electoral quota; the choice of quota affects the properties of the , with smaller quotas leaving fewer seats left over for small parties to pick up, and larger quotas leaving more seats.

The two most common quotas 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. LR-Hare is sometimes called Hamilton's method, named after Alexander Hamilton, who devised the method in 1792.^[4]

The Droop quota is given by:

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

and is applied to elections in South Africa.

The Hare quota is more generous to less popular parties and the Droop quota to more popular parties. Specifically, the Hare quota is unbiased in the number of seats it hands out, and so is more proportional than the Droop quota (which tends to be biased towards larger parties).^[5][6][7][8]

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

Hare quota

Droop quota

Pros and cons

It is easy for a voter to understand how the largest remainder method allocates seats. The Hare quota gives no advantage to larger or smaller parties, while the Droop quota is biased in favor of larger parties.^[9] However, in small legislatures with no threshold, the Hare quota can be manipulated by running candidates on many small lists, allowing each list to pick up a single remainder seat.^[10]

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 so-called Alabama paradox). The highest averages methods avoid this latter paradox, though at the cost of quota violation.^[11]

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