Largest remainder method

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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 or the Hare-Niemeyer method, and is the third-most common apportionment rule worldwide (after the d'Hondt and Sainte-Laguë highest averages methods). 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).