A largest-remainder quota method used for US Congressional apportionment from 1850 to 1900. Also known as the Hamilton method or the Vinton method. Equivalent to the largest remainder method using the Hare quota.
Arguments
- size
An integer number of seats to apportion across units, or a vector of numbers of seats, one for each column of
pop. Must be non-negative.- pop
A vector or matrix of population sizes or proportions for each unit. If a matrix is provided, the apportionment algorithm is applied columnwise: each row is a unit and each column is a replicate. For example, with congressional apportionment, the matrix would have 50 rows and as many columns as hypothetical census population scenarios.
Value
An integer vector or matrix of the same dimensions as pop, containing the
number of seats apportioned to each unit.
Details
The Hamilton-Vinton method first computes the exact quota for each unit:
$$q_i = \frac{p_i \cdot H}{P}$$
where \(p_i\) is the population of unit \(i\), \(H\) is the total
number of seats (size), and \(P = \sum_i p_i\) is the total population.
Each unit initially receives \(\lfloor q_i \rfloor\) seats. Any remaining
seats are awarded one at a time to the units with the largest fractional
remainders \(q_i - \lfloor q_i \rfloor\).
The method satisfies the quota property: no unit ever receives fewer than \(\lfloor q_i \rfloor\) or more than \(\lceil q_i \rceil\) seats. However, it is susceptible to the "Alabama paradox," in which increasing the total house size can paradoxically cause a unit to lose a seat.