Largest remainders method

The largest remainder methods or quota methods are methods of allocating seats proportionally that are based on calculating a quota, i.e. a certain number of votes needed to be guaranteed a seat in parliament. Then, any leftover seats are handed over to "plurality" winners (the parties with the largest remainders, i.e. the most "leftover" votes). They are typically contrasted with the more popular highest averages methods (also called divisor methods).

Divisor methods are generally preferred by social choice theorists to the largest remainder methods because they are less susceptible to apportionment paradoxes. In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats. Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically. Divisor methods also satisfy resource or house monotonicity, which says that increasing the number of seats in a legislature should not cause a state to lose a seat (a situation known as an Alabama paradox).

When using the Hare quota, the method is known as the Hare–Niemeyer or Hamilton method.

Method
The largest remainder methods require the numbers of votes for each party to be divided by a quota representing the number of votes required to win 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 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.

Largest remainder methods can also be used to apportion votes among solid coalitions, as in the case of the single transferable vote, which becomes the largest-remainders method when voters are all partisans (i.e. only rank candidates of their own party).

Quotas
There are several possible choices for the electoral quota; the choice of quota affects the properties of the corresponding largest remainder method, with smaller quotas leaving fewer seats left over for small parties to pick up, and larger quotas leaving more seats. As a result, a larger quota is, somewhat counterintuitively, always more favorable to smaller parties.

The two most common quotas are the Hare quota and the Droop quota. The use of a particular quota with one of the largest remainder methods is often abbreviated as "LR-[quota name]", such as "LR-Droop".

The Hare (or simple) quota is defined as follows:


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

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, 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.

The Droop quota is given by:
 * $$\frac{\text{total votes}}{\text{total seats}+1}$$

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).

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

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. 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.

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 Alabama paradox). The highest averages methods avoid this latter paradox, though at the cost of very rare quota violations.

Technical evaluation and paradoxes
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, this comes at the cost of paradoxical behavior. The Alabama paradox is when an increase in the total number of seats 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: