Quota method

The quota or divide-and-rank methods make up a category of apportionment rules, i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g. parties or federal states). The quota methods begin by calculating an entitlement (basic number of seats) for each party, by dividing their vote totals by an electoral quota (a fixed number of votes needed to win a seat, as a unit). Then, leftover seats, if any are allocated by rounding up the apportionment for some parties. These rules are typically contrasted with the more popular highest averages methods (also called divisor methods).

By far the most common quota method are the largest remainders or quota-shift methods, which assign any leftover seats to the "plurality" winners (the parties with the largest remainders, i.e. most leftover votes).

When using the Hare quota, this rule is called Hamilton's method, and is the third-most common apportionment rule worldwide (after Jefferson's method and Webster's method).

Despite their intuitive definition, quota methods are generally disfavored by social choice theorists as a result of apportionment paradoxes. In particular, the largest remainder methods exhibit the no-show paradox, i.e. voting for a party can cause it to lose seats. The largest remainders methods are also vulnerable to spoiler effects and can fail resource or house monotonicity, which says that increasing the number of seats in a legislature should not cause a party to lose a seat (a situation known as an Alabama paradox).

The largest remainder method divides each party's vote total by a quota. Usually, quota is derived by dividing the number of valid votes cast, 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.

Largest remainder methods produces similar results to single transferable vote or the quota Borda system, where voters organize themselves into solid coalitions. The single transferable vote or the quota Borda system behave like the largest-remainders method when voters all behave like strict partisans (i.e. only mark preferences for candidates of one party).

There are several possible choices for the electoral quota. The choice of quota affects the properties of the corresponding largest remainder method, and particularly the seat bias. Smaller quotas allow small parties to pick up seats, while larger quotas leave behind more votes. A somewhat counterintuitive result of this is that a larger quota will always be more favorable to smaller parties. A party hoping to win multiple seats sees fewer votes captured by a single popular candidate when the quota is small.

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".