largestremainder

Largest remainder method, also called the Hamilton method or Hare method, is a proportional apportionment procedure used to distribute a fixed number of seats among groups in proportion to a size measure such as population. The method computes standard quotas q_i = (population_i / sum population) total_seats. Each group is initially allocated floor(q_i) seats. The remaining seats equal total_seats minus the sum of the floors. Remainders r_i = q_i - floor(q_i) are computed and the leftover seats are assigned to groups with the largest remainders in order until all seats are allocated. In case of ties, predefined rules determine the outcome.

Example: three groups with populations 50, 30, 20; total population 100; total seats 7. Quotas: 3.5, 2.1, 1.4. Floors: 3, 2, 1; remaining seats: 1. Remainders: 0.5, 0.1, 0.4. The remaining seat goes to the group with 0.5, giving final allocation 4, 2, 1.

Properties and caveats: The largest remainder method satisfies the quota rule, in that each allocation is either floor or ceil of its quota. It is simple to implement but can produce paradoxes such as the Alabama paradox, where increasing the total number of seats can cause a group to lose a seat. Ties for remainders may require special rules.

History and usage: Also known as the Hamilton method or Hare method, named after Alexander Hamilton or Thomas Hare. It was widely used in the 19th and early 20th centuries in various legislatures; in modern practice many jurisdictions prefer divisor methods, which have different incentive and monotonicity properties.