A divisor method that uses ceiling (round-up) rounding, proposed by John Quincy Adams. It was never adopted for US Congressional apportionment.
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 Adams method finds a common divisor \(d\) such that ceiling-rounded
quotients sum to the desired house size:
$$\sum_{i} \left\lceil \frac{p_i}{d} \right\rceil = H$$
where \(p_i\) is the population of unit \(i\) and \(H\) is the total
number of seats (size). The divisor \(d\) is adjusted iteratively until
this condition is met.
Because every non-zero fractional remainder is always rounded up, the Adams method is the most generous to small units among the classical divisor methods. Any unit with a positive population receives at least one seat, which can over-represent small states or parties relative to their population share.